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IN  MEMORIAM 
FLORIAN  CAJORI 


PLANE  AND  SPHERICAL 
TRIGONOMETRY 


BY 

ELMER   A.    LYMAN 

MICHIGAN    STATE   NORMAL   COLLEGE 
AND 

EDWIN   C.    GODDARD 

UNIVERSITY   OF  MICHIGAN 


ALLYN    AND    BACON 

Boston  anti  Chicago 


COPYRIGHT,    1899,    1900, 
BY  ELMEK    A.  LYMAN 
AND  EDWIN  C.   GODDARD. 


NortoootJ 

J.  S.  Gushing  &  Co.  -  Berwick  fc  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 


MANY  American  text-books  on  trigonometry  treat  the  solution 
of  triangles  quite  fully  j  English  text-books  elaborate  analytical 
trigonometry;  but  no  book  available  seems  to  meet  both  needs 
adequately.  To  do  that  is  the  first  aim  of  the  present  work,  in 
the  preparation  of  which  nearly  everything  has  been  worked  out 
and  tested  by  the  authors  in  their  classes. 

The  work  entered  upon,  other  features  demanded  attention. 
For  some  unaccountable  reason  nearly  all  books,  in  proving  the 
formulae  for  functions  of  a  ±  /?,  treat  the  same  line  as  both  posi- 
tive and  negative,  thus  vitiating  the  proof ;  and  proofs  given  for 
acute  angles  are  (without  further  discussion)  supposed  to  apply 
to  all  angles,  or  it  is  suggested  that  the  student  can  draw  other 
figures  and  show  that  the  formulae  hold  in  all  cases.  As  a 
matter  of  fact  the  average  student  cannot  show  anything  of  the 
kind ;  and  if  he  could,  the  proof  would  still  apply  only  to  combi- 
nations of  conditions  the  same  as  those  in  the  figures  actually 
drawn.  These  difficulties  are  avoided  by  so  wording  the  proofs 
that  the  language  applies  to  figures  involving  any  angles,  and  to 
avoid  drawing  the  indefinite  number  of  figures  necessary  fully 
to  establish  the  formulae  geometrically,  the  general  case  is  proved 
algebraically  (see  page  58). 

Inverse  functions  are  introduced  early,  and  used  constantly. 
Wherever  computations  are  introduced  they  are  made  by  means 
of  logarithms.  The  average  student,  using  logarithms  for  a  short 
time  and  only  at  the  end  of  the  subject,  straightway  forgets  what 
manner  of  things  they  are.  It  is  hoped,  by  dint  of  much  prac- 
tice, extended  over  as  long  a  time  as  possible,  to  give  the  student 
a  command  of  logarithms  that  will  stay.  The  fundamental  for- 
mulae of  trigonometry  must  be  memorized.  There  is  no  substi- 
tute for  this.  For  this  purpose  oral  work  is  introduced,  and 
there  are  frequent  lists  of  review  problems  involving  all  prin- 
ciples and  formulae  previously  developed.  These  lists  serve  the 

iii 


COPYRIGHT,    1899,    1900, 
BY  ELMER    A.  LYMAN 
AND  EDWIN  C.  GODDARD. 


Nortoooti  -Jjiress 

J.  S.  Gushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 


MANY  American  text-books  on  trigonometry  treat  the  solution 
of  tnangles  quite  fully;  English  text-books  elaborate  analytca" 
trigonometry;  but  no  book  available  seems  to  meet  both  needs 
adequately.  To  do  that  is  the  first  aim  of  the  present  work  in 
the  preparation  of  which  nearly  everything  has  been  worked  'out 
and  tested  by  the  authors  in  their  classes. 

The  work  entered  upon,  other  features  demanded  attention 
For  some  unaccountable  reason  nearly  all  books,  in  proving  the 
formula,  for  functions  of  «  ±  A  treat  the  same  line  as"  both'poS 
bve  and  negative,  thus  vitiating  the  proof;  and  proofs  given  for 
acu  e  anges  are  (without  further  discussion)  supposed  to  apply 
to  all  angles,  or  ,t  is  suggested  that  the  student  can  draw  oXr 
hgures  and  show  that  the  formulae  hold  in  all  cases  A?  , 
matter  of  fact  the  average  student  cannot  show  anything  of  he 
kind;  and  lf  he  could,  the  proof  would  still  apply  only  to  comb' 
nations  of  conditions  the  same  as  those  in  the  figures  ac  Til 

These  difficuies  ar 


In,,™  oo,  „,      toraa  ,„, 

Wh«,,v,,  «ompuMlol,,  „  lnlloduoed  „, 


££TO 

there  are  frequent  lists  of  review  problems  involving  all  prin- 
ciples  and  formulae  previously  developed.     These  lists  serve  the 

iii 


iv  PREFACE. 

further  purpose  of  throwing  the  student  on  his  own  resources, 
and  compelling  him  to  find  in  the  problem  itself,  and  not  in  any 
model  solution,  the  key  to  its  solution,  thus  developing  power, 
instead  of  ability  to  imitate.  To  the  same  end,  in  the  solution 
of  triangles,  divisions  and  subdivisions  into  cases  are  abandoned, 
and  the  student  is  thrown  on  his  own  judgment  to  determine 
which  of  the  three  possible  sets  of  formulae  will  lead  to  the  solu- 
tions with  the  data  given.  Long  experience  justifies  this  as 
clearer  and  simpler.  The  use  of  checks  is  insisted  upon  in  all 
computations. 

For  the  usual  course  in  plane  trigonometry  Chapters  I-VII, 
omitting  Arts.  26,  27,  contain  enough.  Articles  marked  *  (as 
Art.  *  26)  may  be  omitted  unless  the  teacher  finds  time  for  them 
without  neglecting  the  rest  of  the  work.  Classes  that  can  accom- 
plish more  will  find  a  most  interesting  field  opened  in  the  other 
chapters.  More  problems  are  provided  than  any  student  is  ex- 
pected to  solve,  in  order  that  different  selections  may  be  assigned 
to  different  students,  or  to  classes  in  different  years.  Do  not 
assign  work  too  fast.  Make  sure  the  student  has  memorized  and 
can  use  each  preceding  formula,  before  taking  up  new  ones. 

No  complete  acknowledgment  of  help  received  could  here  be 
made.  The  authors  are  under  obligation  to  many  for  general 
hints,  and  to  several  who,  after  going  over  the  proof  with  care, 
have  given  valuable  suggestions.  The  standard  works  of  Levett 
and  Davison,  Hobson,  Henrici  and  Treutlein,  and  others  have 
been  freely  consulted,  and  while  many  of  the  problems  have  been 
prepared  by  the  authors  in  their  class-room  work,  they  have  not 
hesitated  to  take,  from  such  standard  collections  as  writers  gen- 
erally have  drawn  upon,  any  problems  that  seemed  better  adapted 
than  others  to  the  work.  Quality  has  not  been  knowingly  sacri- 
ficed to  originality.  Corrections  and  suggestions  will  be  gladly 
received  at  any  time. 

E.  A.  L.,  YPSILANTI. 

E.  C.  G.,  ANN  ARBOR. 
October,  1900. 


CONTENTS. 

CHAPTER  I.  ANGLES  —  MEASUREMENT  OF  ANGLES. 

PAGE 

Angles ;  magnitude  of  angles 1 

Rectangular  axes ;  direction       ...         .^      ....  2 

Measurement;  sexagesimal  and  circular  systems  of  measurement; 

the  radian     ...........  3 

Examples 6 

CHAPTER  II.     THE  TRIGONOMETRIC  FUNCTIONS. 

Function  defined 8 

The  trigonometric  functions 9 

Fundamental  relations        .........  11 

Examples    ............  14 

Functions  of  0°,  30°,  45°,  60°,  90° 15 

Examples 18 

Variations  in  the  trigonometric  functions 19 

Graphic  representation  of  functions 22 

Examples 27 

CHAPTER  III.  FUNCTIONS  OF  ANY  ANGLE  —  INVERSE 
FUNCTIONS. 

Relations  of  functions  of    -  0,  90°  ±  0,  180°  ±  0,  270°  ±  0  to  the 

functions  of  0 29 

—Inverse  functions 35 

Examples 36 

Review 38 

CHAPTER  IV.    COMPUTATION  TABLES. 

Natural  functions 40 

Logarithms          .        .        .        .        .        .        .        .        .        .        .40 

Laws  of  logarithms     ..........  42 

Use  of  tables 45 

Cologarithms 49 

Examples 50 


CONTENTS. 


CHAPTER  V.     APPLICATIONS. 

PAGE 

Measurements  of  heights  and  distances 51 

Common  problems  in  measurement    .......  52 

Examples 54 

CHAPTER  VI.     GENERAL  FORMULAE  —  TRIGONOMETRIC 
EQUATIONS  AND  IDENTITIES. 

Sine,  cosine,  tangent  of  a  ±  ft .56 

Examples 59 

Sin  0  ±  sin  <£,  0  ±  cos  <f> .        .  61 

Examples 62 

Functions  of  the  double  angle 63 

Functions  of  the  half  angle 64 

Examples 64 

Trigonometric  equations  and  identities 66 

Method  of  attack 66 

Examples 67 

Simultaneous  trigonometric  equations 69 

Examples    ............  70 

CHAPTER  VII.     TRIANGLES. 

Laws  of  sines,  tangents,  and  cosines 72 

Area  of  the  triangle 76 

Solution  of  triangles 76 

Ambiguous  case 78 

Model  solutions 80 

Examples 83 

Applications 84 

Review 86 

CHAPTER  VIII.    MISCELLANEOUS. 

Incircle,  circumcircle,  escribed  circle 92 

Orthocentre,  centroid,  medians 94 

Examples 96 

CHAPTER  IX.     SERIES. 

Exponential  series 97 

Logarithmic  series       ..........  99 

Computation  of  logarithms^ 100 

Be  Moivre's  theorem 103 

Computation  of  natural  functions 104 

Hyperbolic  functions  .         . 109 

Examples 110 


CONTENTS.  vii 

• 
CHAPTER  X.     SPHERICAL  TRIGONOMETRY. 

PAGE 

Spherical  triangles      .        .         .        .         .        .        .        .        .        .112 

General  formulae          .         .        .         .        .        .        .        .        .         .114 

Right  spherical  triangles ."    123 

Area  of  spherical  triangles 125 

Examples 128 

CHAPTER  XI.     SOLUTION  OF  SPHERICAL  TRIANGLES. 

General  principles 129 

Formulae  for  solution 130 

Model  solutions 131 

Ambiguous  cases 132 

Right  triangles 134 

Species 135 

Examples    ............  137 

Applications  to  Geodesy  and  Astronomy 138 


PLANE   TRIGONOMETRY. 

CHAPTER   I. 

ANGLES  — MEASUREMENT   OF   ANGLES. 

1.  Angles.  It  is  difficult,  if  not  impossible,  to  define  an 
angle.  This  difficulty  may  be  avoided  by  telling  how  it 
is  formed.  If  a  line  revolve  about  one  of  its  points,  an  angle 
is  generated,  the  magnitude  of  the  angle  depending  on  the 
amount  of  the  rotation. 

Thus,  if  one  side  of  the  angle  6,  as  OR,  be  originally  in 
the  position  OX,  and  be  revolved  about  the  point  0  to  the 
position  in  the  figure,  the 
angle  XOR  is  generated. 
OX  is  called  the  initial  line, 
and  any  position  of  OR  the 

terminal  line  of  the  angle  XX^ \L x 

formed.      The   angle   9   is      ,   /*' 
considered  positive  if  gener-     •''' 
ated  by  a  counter-clockwise 

rotation  of  OR,  and  hence  negative  if  generated  by  a  clockwise 
rotation.  The  magnitude  of  6  depends  on  the  amount  of 
rotation  of  OR,  and  since  the  amount  of  such  rotation  may 
be  unlimited,  there  is  no  limit  to  the  possible  magnitude  of 
angles,  for,  evidently,  the  revolving  line  may  reach  the  posi- 
tion OR  by  rotation  through  an  acute  angle  0,  and,  likewise, 
by  rotation  through  once,  twice,  •••,  n  times  360°,  plus  the 
acute  angle  0.  So  that  XOR  may  mean  the  acute  angle 

e,  e  +  360°,  e  +  720°,  •••,  e  +  n  •  360°. 


2  PLANE   TRIGONOMETRY. 

In  reading  an  angle,  read  first  the  initial  line,  then  the 
terminal  line.  Thus  in  the  figure  the  acute  angle  XOR,  or 
xr,  is  a  positive  angle,  and  ROX,  or  rx,  an  equal  negative 
angle. 

Ex.  1.  Show  that  if  the  initial  lines  for  £,  f,  -2^,  —  |,  right  angles  are 
the  same,  the  terminal  lines  may  coincide. 

2.  Name  four  other  angles  having  the  same  initial  and  terminal  lines 
as  j  of  a  right  angle ;  as  f  of  a  right  angle ;  as  f  of  a  right  angle. 

2.  Rectangular  axes.     Any  plane  surface  may  be  divided 
by  two  perpendicular  straight  lines  XX'  and  YY'  into  four 

portions,  or  quadrants. 

XXf  is  known  as  the  x-axis, 
YY'  as  the  y-axis,  and  the  two 
together  are  called  axes  of  refer- 

X     ence.     Their  intersection  0  is  the 

origin,  and   the   four   portions   of 
the  plane   surface,  XOY,    YOX', 
X'OY',  Y'OX,  are  called  respec- 
FlG  2  tively  the  first,  second,  third,  and 

fourth  quadrants.     The  position  of 

any  point  in  the  plane  is  determined  when  we  know  its  dis- 
tances and  directions  from  the  axes. 

3.  Any  direction  may  be  considered  positive.     Then  the 
opposite  direction  must  be  negative.    Thus,  if  AB  represents 
any  positive  line,  BA  is  an  equal  nega- 
tive   line.       Mathematicians    usually 

consider  lines  measured  in  the  same  direction  as  OX  or  OY 
(Fig.  2)  as  positive.  Then  lines  measured  in  the  same  direc- 
tion as  OX'  or  OY'  must  be  negative. 

The  distance  of  any  point  from  the  «/-axis  is  called  the 
abscissa,  its  distance  from  the  #-axis  the  ordinate,  of  that 
point ;  the  two  together  are  the  coordinates  of  the  point, 
usually  denoted  by  the  letters  x  and  y  respectively,  and 
written  (x,  y). 


ANGLES  —  MEASUREMENT. 


Y 

p' 

P 

P" 

0 

N        X 

Y' 

FIG 


When  taken  with  their  proper  signs,  the  coordinates  define  completely 
the  position  of  the  point.  Thus,  if  the  point  P  is  +  a  units  from  YY', 
and  -f  b  units  from  XX',  any  convenient 
unit  of  length  being  chosen,  the  position  of 
P  is  known.  For  we  have  only  to  measure 
a  distance  ON  equal  to  a  units  along  OX, 
and  then  from  N  measure  a  distance  b 
units  parallel  to  OY,  and  we  arrive  at  the 
position  of  the  point  P,  (a,  &).  In  like 
manner  we  may  locate  P',  (—  a,  b),  in  the 
second  quadrant,  P",  (—a,  —  &),  in  the 
third  quadrant,  and  P'",  (a,  —  b),  in 
the  fourth  quadrant. 

Ex.   Locate  (2,  -2);  (0,0);  (-8,  -7);  (0,  5);  (-2,  0);  (2,  2); 
(m,  n). 

4.  If  OX  is  the  initial  line,  0  is  said  to  be  an  angle  of  the 
first,  second,  third,  or  fourth  quadrant,  according  as  its  ter- 
minal line  is  in  the  first,  second,  third,  or  fourth  quadrant. 
It  is  clear  that  as  OR  rotates  its  quality  is  in  no  way  affected, 
and  hence  it  is  in  all  positions  considered  positive,  and  its  ex- 
tension through  0,  OR',  negative. 

The  student  should  notice  that  the  initial  line  may  take  any  position 
and  revolve  in  either  direction.  While  it  is  customary  to  consider  the 
counter-clockwise  rotation  as  forming  a  positive  angle,  yet  the  condi- 
tions of  a  figure  may  be  such 
that  a  positive  angle  may  be 
generated  by  a  clockwise  rota- 
tion. Thus  the  angle  XOR  in 
each  figure  may  be  traced  as 
a  positive  angle  by  revolving 
the  initial  line  OX  to  the  posi- 
tion OR.  No  confusion  can  result  if  the  fact  is  clear  that  when  an 
angle  is  read  XOR,  OX  is  considered  a  positive  line  revolving  to  the 
position  OR.  OX'  and  OR'  then  are  negative  lines  in  whatever  direc- 
tions drawn.  These  conceptions  are  mere  matters  of  agreement,  and  the 
agreement  may  be  determined  in  a  particular  case  by  the  conditions  of 
the  problem  quite  as  well  as  by  such  general  agreements  of  mathema- 
ticians as  those  referred  to  in  Arts.  3  and  4  above. 

5.  Measurement.     All   measurements  are  made  in  terms 
of  some  fixed  standard  adopted  as  a  unit.     This  unit  must 


R  x  x 
V/  | 
Xf  y 


x' 


4  PLANE   TRIGONOMETRY. 

be  of  the  same  kind  as  the  quantity  measured.  Thus,  length 
is  measured  in  terms  of  a  unit  length,  surface  in  terms  of  a 
unit  surface,  weight  in  terms  of  a  unit  weight,  value  in  terms 
of  a  unit  value,  an  angle  in  terms  of  a  unit  angle. 

The  measure  of  a  given  quantity  is  the  number  of  times  it 
contains  the  unit  selected. 

Thus  the  area  of  a  given  surface  in  square  feet  is  the 
number  of  times  it  contains  the  unit  surface  1  sq.  ft.  ;  the 
length  of  a  road  in  miles,  the  number  of  times  it  contains 
the  unit  length  1  mi. ;  the  weight  of  a  cargo  of  iron  ore  in 
tons,  the  number  of  times  it  contains  the  unit  weight  1  ton ; 
the  value  of  an  estate,  the  number  of  times  it  contains  the 
unit  value  $1. 

The  same  quantity  may  have  different  measures,  according 
to  the  unit  chosen.  So  the  measure  of  80  acres,  when  the 
unit  surface  is  1  acre,  is  80,  when  the  unit  surface  is  1  sq.  rd., 
is  12,800,  when  the  unit  surface  is  1  sq.  yd.,  is  387,200. 
What  is  its  measure  in  square  feet  ? 

6.  The  essentials  of  a  good  unit  of  measure  are  : 

1.  That  it  be  invariable,  i.e.  under  all  conditions  bearing 
the  same  ratio  to  equal  magnitudes. 

2.  That  it  be  convenient  for  practical  or  theoretical  pur- 
poses. 

3.  That  it  be  of  the  same  kind  as  the  quantity  measured. 

7.  Two  systems  of  measuring  angles  are  in  use,  the  sexa- 
gesimal and  the  circular. 

The  sexagesimal  system  is  used  in  most  practical  applica- 
tions. The  right  angle,  the  unit  of  measure  in  geometry, 
though  it  is  invariable,  as  a  measure  is  too  large  for  con- 
venience. Accordingly  it  is  divided  into  90  equal  parts, 
called  degrees.  The  degree  is  divided  into  60  minutes,  and 
the  minute  into  60  seconds.  Degrees,  minutes,  seconds,  are 
indicated  by  the  marks  °  '  ",  as  36°  20'  15". 

The  division  of  a  right  angle  into  hundredths,  with  subdivisions  into 
hundredths,  would  be  more  convenient.  The  French  have  proposed  such 


MEASUREMENT   OF   ANGLES. 


a  centesimal  system,  dividing  the  right  angle  into  100  grades,  the  grade 
into  100  minutes,  and  the  minute  into  100  seconds,  marked  8N  NV,  as  50& 
70V  28".  The  great  labor  involved  in  changing  mathematical  tables, 
instruments,  and  records  of  observation  to  the  new  system  has  prevented 
its  adoption. 

8.   The  circular  system  is  important  in  theoretical  con- 
siderations.    It  is  based  on  the  fact  that  for  a  given  angle 
the  ratio  of  the  length  of  its  arc  to  the  length  of  the  radius 
of  that  arc  is  constant,  i.e.  for  a  fixed 
angle  the  ratio  arc  :  radius  is  the  same 
no    matter   what    the   length   of   the 
radius.     In  the  figure,  for  the  angle  0, 

OA       OB       00 


AA     BB'      CO' 

That  this  ratio  of  arc  to  radius  for  a  fixed  angle  is  constant 
follows  from  the  established  geometrical  principles : 

1.  The  circumference  of  any  circle  is  2?r  times  its  radius. 

2.  Angles  at  the  centre  are  in  the  same  ratio  as  their  arcs. 

The  Radian.     It  follows  that  an  angle  whose  arc  is  equal 
in  length  to  the  radius  is  a  constant  angle  for  all  circles, 
since  in  four  right  angles,  or  the  perigon,  there  are  always 
2  TT  such  angles.      This  constant  angle, 
tvhose  arc  is  equal  in  length  to  the  radius, 
is    taken    as  the   unit  angle  of  circular 
measure,  and  is  called  the  radian.    From 
the  definition  we  have 

4  right  angles  =  360°  =  2  TT  radians, 
2  right  angles  =  180°  =  TT  radians, 

1  right  angle    =    90°  =  -  radians. 


FIG.  6. 


TT  is  a  numerical  quantity,  3.14159+,  and  not  an  angle.      When  we 
speak  of  180°  as  TT,  90°  as  £,  etc.,  we  always  mean  IT  radians,  -  radians,  etc. 


6  PLANE   TRIGONOMETRY. 

9.   To  change  from  one  system  of  measurement  to  the 
other  we  use  the  relation, 

2  TT  radians  =  360°. 

1 80° 
.-.  1  radian  =  — =  57°.  2958-; 

7T 

i.e.  the  radian  is  57°.3,  approximately. 

Ex.  1.    Express  in  radians  75°  30'. 

75°  30'  =  75°.5 ;  1  radian  =  57°.3. 


.-.  75°  30'  =  -  -  =  1.317  radians. 
57.3 

2.    Express  in  degree  measure  3.6  radians. 

1  radian  =  57°.3. 
.-.  3.6  radians  =  3.6  x  57°.3  =  206°  16'  48". 

EXAMPLES. 

1.  Construct,  approximately,  the   following   angles :   50°,  -  20°,  90°, 

179°,    -  135°,   400°,    -  380°,  1140°,    -  radians,    -  radians,    -  -  radians, 

43  6 

Q                      19 
3  TT  radians,    — —  radians,   radians.      Of  which  quadrant  is  each 

angle? 

2.  What  is  the  measure  of : 

(a)  £  of  a  right  angle,  when  30°  is  the  unit  of  measure  ? 

(6)  an  acre,  when  a  square  whose  side  is  10  rds.  is  the  unit?  j«  ^ 

(c)  ra  miles,  when  y  yards  is  the  unit?  \Ji 

3.  What  is  the  unit  of  measure,  when  the  measure  of  2|  miles  is  50? 

4.  The  Michigan  Central  R.R.  is  535  miles  long,  and  the  Ann  Arbor 
R.R.  is  292  miles  long.     Express  the  length  of  the  first  in  terms  of  the 
second  as  a  unit. 

5.  What  will  be  the  measure  of  the  radian  when  the  right  angle  is 
taken  for  the  unit  ?     Of  the  right  angle  when  the  radian  is  the  unit  ? 

6.  In  which    quadrant  is  45°?   10°?    -60°?   145°?  1145°?    -725°? 
Express  each  in  right  angles ;  in  radians. 

7.  Express  in  sexagesimal  measure 

-,  — ,  1,  6.28,  i,  — ,   -~,  radians. 
3'  12  TT     3  3  ' 


EXAMPLES.  7 

8.  Express  in  each  system  an  interior  angle  of  a  regular  hexagon  ; 
an  exterior  angle.    /  £,  0°  ~  %  ^ 

9.  Find  the  distance  in  miles  between  two  places  on  the  earth's 
equator  which  are  11°  15'  apart.    (The  earth's  radius  is  about  3963  miles.) 

10.  Find  the  length  of  an  arc  which  subtends  an  angle  of  4  radians 
at  the  centre  of  a  circle  of  radius  12  ft.  3  in.   «*  ct  i«^\  7  */  ;  I 

11.  An  arc  15  yds.  long  contains  3  radians.     Find  the  radius  of  the 
circle. 

12.  Show  that  the  hour  and  minute  hands  of  a  watch  turn  through 
angles  of  30'  and  6°  respectively  per  minute;  also  find  in  degrees  and  in 
radians  the  angle  turned  through  by  the  minute  hand  in  3  hrs.  20 

13.  Find  the  number  of  seconds  in  an  arc  of  1  mile  on  the  equator  ; 
also  the  length  in  miles  of  an  arc  of  1'  (1  knot). 

"   14.   Find  to  three  decimal  places  the  radius  of  a  circle  in  which  the 
arc  of  71°  36'  3".6  is  15  in.  long.  *jf  fifoQl 

15.  Find  the  ratio  of  -  to  5°. 

6 

16.  What  is  the  shortest  distance  measured  on  the  earth's  surface 
from  the  equator  to  Ann  Arbor,  latitude  +  42°  16'  48"?  s  <J  t?,  *-  ^fowo  *f  3  0« 

17.  The  difference  of  two  angles  is  10°,  and  the  circular  measure  of 
their  sum  is  2.     Find  the  circular  measure  of  each  angle. 

18.  A  water  wheel  of  radius  6  ft.  makes  30  revolutions  per  minute. 
Find  the  number  of  miles  per  hour  travelled  by  a  point  on  the  rim. 


CHAPTER   II. 

THE  TRIGONOMETRIC   FUNCTIONS. 

10.  Trigonometry,  as  the  word  indicates,  was  originally 
concerned   with   the    measurement    of    triangles.      It   now 
includes   the   analytical    treatment   of    certain  functions   of 
angles,  as  well  as  the  solution  of  triangles  by  means  of  cer- 
tain relations  between  the  functions  of  the  angles  of  those 
triangles. 

11.  Function.     If  one  quantity  depends  upon  another  for 
its  value,  the  first  is  called  a  function  of  the  second.     It 
always  follows  that  the  second  quantity  is  also  a  function  of 
the  first ;  and,  in  general,  functions  are  so  related  that  if  one 
is  constant  the  other  is  constant,  and  if  either  varies  in  value, 
the  other  varies.     This  relation  may  be  extended  to  any 
number  of  mutually  dependent  quantities. 

Illustration.  If  a  train  moves  at  a  rate  of  30  miles  per 
hour,  the  distance  travelled  is  a  function  of  the  rate  and 
time,  the  time  is  a  function  of  the  rate  and  distance,  and  the 
rate  is  a  function  of  the  time  and  distance. 

Again,  the  circumference  of  a  circle  is  a  function  of  the 
radius,  and  the  radius  of  the  circumference,  for  so  long  as 
either  is  constant  the  other  is  constant,  and  if  either  changes 
in  value,  the  other  changes,  since  circumference  and  radius 
are  connected  by  the  relation  C  —  2  irR. 

Once  more,  in  the  right  triangle 
NOP,  the  ratio  of  any  two  sides  is 
a  function  of  the  angle  a,  because 
all  the  right  triangles  of  which  a  is 
one  angle  are  similar,  i.e.  the  ratio 
8 


THE   TRIGONOMETRIC   FUNCTIONS. 


of  two  corresponding  sides  is  constant  so  long  as  a  is  con- 
stant, and  varies  if  a  varies. 
Thus,  the  ratios 


and 


NP 
OP 

ON 
NP 


N'P1 
OP' 

ON' 


N"P" 
OP" 

ON" 


etc., 


N'P'     N"P"' 
depend  on  a  for  their  values,  i.e.  are  functions  of  a. 

12.  The  trigonometric  functions.  In  trigonometry  six 
functions  of  angles  are  usually  employed,  called  the  trigono- 
metric functions. 

By  definition  these  functions  are  the  six  ratios  between  the 
sides  of  the  triangle  of  reference  of  the  given  angle.  The 
triangle  of  reference  is  formed  by  drawing  from  some  point  in 
the  initial  line,  or  the  initial  line  produced,  a  perpendicular  to 
that  line  meeting  the  terminal  line  of  the  angle. 


0 


X     N 


o 


FIG.  8. 


Let  a  be  an  angle  of  any  quadrant.  Each  triangle  of 
reference  of  a,  NOP,  is  formed  by  drawing  a  perpendicular 
to  OX,  or  OX  produced,  meeting  the  terminal  line  OR  in  P. 


10  PLANE   TRIGONOMETRY. 

If  a  is  greater  than  360°,  its  triangle  of  reference  would 
not  differ  from  one  of  the  above  triangles. 

It  is  perhaps  worthy  of  notice  that  the  triangle  of  reference  might  be 
defined  to  be  the  triangle  formed  by  drawing  a  perpendicular  to  either 
side  of  the  angle,  or  that  side  produced,  meet- 
ing the  other  side  or  the  other  side  produced. 
In  the  figure,  NOP  is  in  all  cases  the  triangle 
.      ^  of  reference  of  «.     The  principles  of  the  fol- 

|    ,.--'jy  lowing  pages  are  the  same  no  matter  which 

s'P  of  the  triangles  is  considered  the  triangle  of 

FIG.  9.  reference.     It  will,  however,  be  as  well,  and 

perhaps  clearer,  to  use  the  triangle   defined 
under  Fig.  8,  and  we  shall  always  draw  the  triangle  as  there  described. 

13.  The  trigonometric  functions  of  a  (Fig.  8)  are  called 
the  sine,  cosine,  tangent,  cotangent,  secant,  and  cosecant  of  «. 
These  are  abbreviated  in  writing  to  sin  a,  cos  a,  tan  a,  cot  a, 
sec  «,  esc  a,  and  are  defined  as  follows  : 

sin  a  =  PpP-  =  ^,  whence  y  =  r  sin  a ; 
cos  a  =  g^  =  —  9  whence  x  =  r  cos  « ; 
tan  a  =  p.frp*  =  — »  whence  y  =  x  tan  a ; 

t)tiS(k  tJO 

cot  a  =    *se  =  —  >  whence  x  =  y  cot  a  • 


sec  a  =      ±  =    »  whence   r  =  x  sec  a  ; 


esc  a  =  —  —  =  —,  whence   r  =  y  esc  a. 
perp.     y 

1  —  cos  a  and  1  —  sin  a,  called  versed-sine  a  and  coversed-sine  a,  respec- 
tively, are  sometimes  used. 

Ex.  1.  Write  the  trigonometric  functions  of  ft,  NPO  (Fig.  8),  and 
compare  with  those  of  a  above. 

The  meaning  of  the  prefix  co  in  cosine,  cotangent,  and  cosecant 
appears  from  the  relations  of  Ex.  1.  For  the  sine  of  an  angle  equals  the 
cosine,  i.e.  the  complement-sine,  of  the  complement  of  that  angle  ;  the  tangent 


THE   TRIGONOMETRIC   FUNCTIONS.  H 

of  an  angle  equals  the  cotangent  of  its  complementary  angle,  and  the  secant 
of  an  angle  equals  the  cosecant  of  its  complement- 
ary angle. 

2.  Express  each  side  of  triangle  ABC  in 
terms  of  another  side,  and  some  function  of  an 
angle  in  all  possible  ways,  as  a  =  b  tan  A,  etc. 

14.  Constancy  of  the  trigonometric  functions.  It  is  impor- 
tant to  notice  why  these  ratios  are  functions  of  the  angle,  i.e. 
are  the  same  for  equal  angles  and  different  for  unequal 
angles.  This  is  shown  by  the  principles  of  similar  triangles. 


\ 

a. 


a 


FIG.  11. 

In  each  figure  show  that  in  all  possible  triangles  of  refer- 
ence for  a  the  ratios  are  the  same,  but  in  the  triangles  of 
reference  for  a  and  a',  respectively,  the  ratios  are  different. 

The  student  must  notice  that  sin  a  is  a  single  symbol.  It  is  the  name 
of  a  number,  or  fraction,  belonging  to  the  angle  a ;  and  if  it  be  at  any 
time  convenient,  we  may  denote  sin  a  by  a  single  letter,  such  as  0,  or  x. 
Also,  sin2  a  is  an  abbreviation  for  (sin  a)2,  i.e.  for  (sin  a)  x  (sin  a) . 
Such  abbreviations  are  used  because  they  are  convenient.  Lock,  Ele- 
mentary Trigonometry. 

15.  Fundamental  relations.  From  the  definitions  of  Art.  13 
the  following  reciprocal  relations  are  apparent : 

sin  a  =  — — ,  csc  a  = > 

csc  a  8ina 

cos  a  =  —  — ,  sec  a  = , 

'c  a  COS  (& 

tan  a  =  -±—,  cot  a 


cot a  tan  a 
Also  from  the  definitions, 

ta_         siii  a  cos  a 

tan  a  = 9  cot  ex,  = • 

cos  a  sin  a 


12  PLANE  TRIGONOMETRY. 

From  the  right  triangle  NOP,  page  9, 

y*  +  3?  =  ia; 
whence  (1)  +     -l, 


oo 


From  (1)  sin2  a + cos2  a  =  1 ;  sin  a  =  VI  —  cos2  a ;  cos  a  =  ? 
(2)  tan2  a  + 1  =  sec2  a ;  tan  ex,  =  Vsee2  a  —  1 ;  se<?  «  =  ? 
(3) 


a— 


The  foregoing  definitions  and  fundamental  relations  are  of 
the  highest  importance,  and  must  be  mastered  at  once.  The 
student  of  trigonometry  is  helpless  without  perfect  familiarity 
with  them. 

These  relations  are  true  for  all  values  of  a,  positive  or  negative,  but 
the  signs  of  the  functions  are  not  in  all  cases  positive,  as  appears  from 
the  fact  that  in  the  triangles  of  reference  in  Fig.  8  x  and  y  are  sometimes 
negative.  The  equations  sin  a=±  Vl  —  cos2  a,  tan  a  =  ±  Vsec2  a  —  1, 
cot  a  =  ±  Vcsc2  a  —  1,  have  the  double  sign  ±.  Which  sign  is  to  be  used 
in  a  given  case  depends  on  the  quadrant  in  which  a  lies. 

16.  The  relations  of  Art.  15  enable  us  to  express  any 
function  in  terms  of  any  other,  or  when  one  function  is 
given,  to  find  all  the  others. 

Ex.  1.    To  express  the  other  functions  in  terms  of  tangent  : 


sn  «= 


sec  a     Vl  +  tan2  a 
tan  a  =  tan  ot ; 


tan  a 


THE   TRIGONOMETRIC   FUNCTIONS. 


13 


In  like  manner  determine  the  relations  to  complete  the  following 
table : 


sin  a 

cosa 

tana 

cot  a 

sec  a 

esc  a 

sin  a 

cos  a 
tana 
cot  a 
sec  a 
esc  a 

4u 
•a,- 

vT- 

._><< 

tan  a 

i 

\|S2S 

t-iXT~eJ» 
\l  |-V  C-c**^. 

;. 

^  i  __ 

e^i^t-^_    ^ 

C  1;      i 

VI  +  tan2  a 
1 
VI  +  tan2  a 
tan  a 

1 

tan  a 

VI  -1-  tan2  a 

VI  +  tan2  a 

tan  a 

2.    Given  sin  a  : 
cos  a  =A 

=  f  ;  find  the  other  funct 

ions. 

3                  H 

^V7      V7 

/I  —  T9£  =  \  V7  ;   tan  a  = 

:  A  A/7  :     SPf»  ft—  —          — 

|V7 


V7 


3.   Given  tan  <£  +  cot  <f>  =  2  ;  find  sin  <£. 


=  2,   tan2  d>  -  2  tan  c!)  +  1  =  0,   tan  <f>  =  1. 


tan 


'1  +  tan2  (j> 

Or,  expressing  in  terms  of  sine  directly, 

cos 

sin2  <J>  +  cos2  </>  =  2  sin  <£  cos  ^>,   sin2  <£  —  2  sin  <f>  cos  <^>  +  cos2  <f>  =  0 ; 
whence        sin  <£  —  cos  <f>  =  0,   sin  <£  =  cos  <£.     .•.  sin  <f>  =  |\/2. 

4.  Prove  sec4  x  —  sec2  a:  =  tan2  x  +  tan4  #. 

sec4  a;  —  sec2  a;  =  sec2  x  (sec2  x  —  !)  =  (!  +  tan2  a:)  tan2  a:  =  tan2  a:  +  tan4  a;. 

5.  Prove  sin6  y  +  cos6  y  =  1  —  3  sin2  y  cos2  y. 

sin6  y  +  cos6  y  =  (sin2  y  +  cos2  #)  (sin4  y  —  sin2  ?/  cos2  y  -f  cos4  #) 

=  (sin2  y  +  cos2  y)2  —  3  sin2  ?/  cos2  y  =  1  —  3  sin2  y  cos2  y. 


PLANE   TRIGONOMETRY. 


+      Cot*     =  sec2csc2 
1  —  cot  z     1  —  tan  z 


tan  3 


cos  2  (sin  2  —  cos  2)      sin  2  (cos  2  —  sin  2) 

sin8  2  —  cos3  2  _  sin2  2  +  sin  2  cos  2  +  cos2  2 

sin  2  cos  2  (sin  2  —  cos  2)  sin  2  cos  2 


sm  2  cos  z         sm  2  cos  2 

In  solving  problems  like  3,  4,  5,  and  6  above,  it  is  usually  safe,  if  no 
other  step  suggests  itself,  to  express  all  other  functions  of  one  member 
in  terms  of  sine  and  cosine.  The  resulting  expression  may  then  be  re- 
duced by  the  principles  of  algebra  to  the  expression  in  the  other  member 
of  the  equation.  For  further  suggestions  as  to  the  solution  of  trigono- 
metric equations  and  identities  see  page  66. 

EXAMPLES. 

1.  Find  the  values  of  all  the  functions  of  a,  if  sin  a  =  f  ;  if  tan  a  =  f  ; 
if  sec  a  =  2  ;  if  cos  a  =  £\/3  ;  if  cot  a  =  \  ;  if  esc  a  =  V2. 

2.  Compute  the  functions  of  each  acute  angle  in  the  right  triangles 
whose  sides  are  :  /I)  3,  4,  5;   (2)  8,  15,  17;   (3)  480,  31,  481  ;   (4)  a,  b,  c; 

„,    2  xy      x**+W 

(5)  -  2L,    _       JL,  x  +  y. 

x-y      x-y 

3.  If  cos  a  =  A,  find  the  value  of  sma  +  tan" 

cos  «  —  cot  a 

4.  If  2  cos  a  =  2  —  sin  a,  find  tan  a. 

5.  If  sec2  «  esc2  a  -  4  =  0,  find  cot  a. 

6.  Solve  for  sin  £  in      13  sin  ft  +  5  cos2  ft  =  11. 
Prove 

"    7.   sin4  <f>  -  cos4  <ft  =  1  -  2  cos2  <£. 

8.  (sin  a  +  cos  a)  (sin  a  —  cos  a)  =  2  sin2  a  —  1. 

9.  (sec  a  +  tan  a)  (sec  «  —  tan  a)  =  1. 

10.    cos2£  (sec2£  -  2  sin2  /?)  =  cos4  J3  +  sin4  0. 

COS  V 

—  sir 
12         sin  w     _  1  +  cos  w 

1  —  cos  w         sin  w 
13. 


FUNCTIONS  OF   CERTAIN   ANGLES.  15 

14.    sin4  t  -  sin2 1  -  cos4  t  -  cos2  t. 

=  sec2  (3  (esc  B  +  1). 
sin  /} 

16.  (tan  A  +  cot^l)2  =  sec2  A  esc2  A. 

17.  sec2  a:  —  sin2  a;  =  tan2  x  +  cos2  a:. 

In  the  triangle  ABC,  right  angled  at  C, 

18.  Given  cos  A  =  T87,  BC  =  45,  find  tan  5,  and  ,4£. 

19.  If  cos  4  =  ™2  ~~  n2,  and  AB  =  m2  +  n2  find  A  C  and  £C. 

m2  +  n2 

20.  If  A  C  =  m  +  n,  BC  =  m  —  n,  find  sin  A,  cos  J5. 

21.  In     examples     18,    19,    20,    above,    prove     sin2  A  -f  cos2  A  =  1 ; 
1  +  tan2  4  =  sec2  4. 

17.  Functions  of  certain  angles.     The  trigonometric  func- 
tions are  numerical  quantities  which  may  be  determined  for 
any  angle.     In  general  these  values  are  taken  from  tables 
prepared  for  the  purpose,  but  the  principles  already  studied 
enable  us  to  calculate  the  functions  of  the  following  angles. 

18.  Functions  of  0°.     If   a  be   a  very  small   angle,   the 
value  of  y  is  very  small,  and 

decreases   as     a    diminishes. 

Clearly,  when   a   approaches  F     12  N 

0°  as  a  limit,  y  likewise  ap- 
proaches 0,  and  x  approaches  r,  so  that  when  a  =  0°, 

y  —  0,  and  x  =  r. 
...  rinO'  =  J  =  0,  ^o°  =  ir^  =  oo, 

co8Q°  =  -=l,  secQ°  = 

r  cos  0° 


In  the  figure  of  Art.  18,  by  diminishing  a  it  is  clear  that  we  can  make 
y  as  small  as  we  please,  and  by  making  a  small  enough,  we  can  make  the 
value  of  y  less  than  any  assignable  quantity,  however  small,  so  that  sin  a  ap- 
proaches as  a  limit  0.  This  is  what  we  mean  when  we  say  sin  0°  =  0. 
In  like  manner,  it  is  evident  that,  by  sufficiently  diminishing  a  we  can 
make  cot  a  greater  than  any  assignable  quantity.  This  we  express  by 
saying  cotO°  =  <x>. 


16 


PLANE  TRIGONOMETRY. 


19.   Functions  of  30°.     Let  NOP  be  the  triangle  of  refer- 

ence  for  an  angle  of  30°.  Make 
triangle  NOP1  =  NOP.  Then 
POP'  is  an  equilateral  triangle 
_x  (why?),  and  ON  bisects  PPf. 
Hence 


FIG.  13. 


C(?s  30°  =  -  = 


Also  x  =  v  ?*2  —  yp  = 

esc  30°  =  2, 


=  -  =          =  i  V3, 
r        2  ?/ 


sec  30°  =  *  V3, 


ton  30°  =  ?  =  -^-=  =  -i  =  iV3,         co*  30°  =  VS. 

V3 


20.   Functions  of  45°.     Let  NOP  be  the  triangle  of  refer- 
ence.    If  angle  NOP  =  45°,  OPN=  45°. 


o 


R 


*  AT 

FIG.  14. 

.  •.  y  =  x,  and  r  =  V#2  +  j/2  =  V2  z?  = 


Then 


ICO  ^/  ^ 

tow  45°  =  -  =  - 
x     x 

Find  cot  45°,  sec  45°,  CM  45°. 


FUNCTIONS  OF   CERTAIN  ANGLES. 


17 


21.  Functions  of  60°.  The  functions  of  60°  may  be  com- 
puted by  means  of  the  figure,  or 
they  may  be  written  from  the  func- 
tions of  the  complement,  or  30°. 
Let  the  student  in  both  ways  show 
that 

sin  60°  =  l  V3,      cos  60°  =  J, 


o 


N  P' 

FIG.  15. 


Compute  also  the  other  functions  of  60°. 


Y 


22.  Functions  of  90°.  If  «  be  an  angle  very  near  90°, 
the  value  of  x  is  very  small,  and  de- 
creases as  a  increases  toward  90°. 
Clearly  when  a  approaches  90°  as  a 
limit,  x  approaches  0,  and  y  ap- 
proaches r,  so  that  when 

x  a  =90°,    z=0,    y  =  r. 


FIG.  16. 


.  •.  sin  90°  =  1,  cos  90°  =  0,  tan  90°  =  oo . 


Compute  the  other  functions.  Also  find  the  functions  of 
90°  from  those  of  its  complement,  0°. 

23.  It  is  of  great  convenience  to  the  student  to  remember 
the  functions  of  these  angles.  They  are  easily  found  by 
recalling  the  relative  values  of  the  sides  of  the  triangles  of 
reference  for  the  respective  angles,  or  the  values  of  the  other 
functions  may  readily  be  computed  by  means  of  the  funda- 
mental relations,  if  the  values  of  the  sine  and  cosine  are 
remembered,  as  follows  : 


« 

0° 

30° 

45° 

60° 

90° 

sine 

Wo 

M 

W2 

iV3 

iVi 

cosine 

W4 

£V3 

W2 

IV! 

iVo 

18  PLANE  TRIGONOMETRY. 

ORAL  WORK. 

1.  Which  is  greater,  sin  45°  or  |  sin  90D  ?  sin  60°  or  2  sin  30°  ? 

2.  From  the  functions  of  60°,  find  those  of  30° ;  from  the  functions  of 
90°,  those  of  0°.     Why  are  the  functions  of  45°  equal  to  the  co-functions 
of  45°? 

3.  Given  sin  A  =  $,  find  cos  A  ;  tan  A. 

4.  Show  that  sin  B  esc  B  =  1 ;  cos  C  sec  C  =  1 ;  cot  x  tan  x  =  1. 

5.  Show  that  sec2  0  -  tan2  0  =  esc2  0  -  cot2  B  -  sin2  0  +  cos2  6. 

6.  Show  that  tan  30°  tan  60°  =  cot  60°  cot  30°  =  tan  45°. 

7.  Show  that  tan  60°  sin2  45°  =  cos  30°  sin  90°. 

8.  Show  that  cos  a  tan  a  =  sin  a ;  sin  ft  cot  ft  =  cos  (3. 

9.  Show  that  *""tan'!??!  -  cos  60°  =  \  cos  0°. 

1  +  tan^oO 

10.    Show  that  (tan  y  +  cot  y)  sin  ycosy  =  1. 

EXAMPLES. 

1.  Show  that  sin  30°  cos  60°  +  cos  30°  sin  60°  =  sin  90°. 

2.  Show  that  cos  60°  cos  30°  +  sin  60°  sin  30°  =  cos  30°. 

3.  Show  that  sin  45°  cos  0°  -  cos  45°  sin  0°  =  cos  45°. 

4.  Show  that  cos2  45°  -  sin2  45°  =  cos  90°. 

5.  Show  that    tan  45°  + tan  0°  = 

1  -  tan  45°  tan  0° 

If  A  =  60°,  verify 


If  a  =  0°,  ft  =  30°,  y  =  45°,  8  =  60°,  c  =  90°,  find  the  values  of 
9.   sin  ft  +  cos  8. 

10.  cos  ft  +  tan  8. 

11.  sin  ft  cos  8  +  cos  ft  sin  8  —  sin  e. 

12.  (sin  ft  +  sin  e)  (cos  a  +  cos  8)  -  4  sin  a  (cos  y  +  sin  e) . 


VARIATIONS  IN   THE  FUNCTIONS. 


19 


24.  Variations  in  the  trigonometric  functions. 

Signs.  Thus  far  no  account  has  been  taken  of  the  signs  of 
the  functions.  By  the  definitions  it  appears  that  these  de- 
pend on  the  signs  of  x,  y,  and  r.  Now  r  is  always  positive, 
and  from  the  figures  it  is  seen  that  x  is  positive  in  the  first 


Sin.  + 
Csc.  + 


•Tan.  •+ 
Cot.  + 


Sin.  + 
Cos.  + 
Tan.  + 
Cot.  + 
Sec.  + 
Csc.  + 


Cos.  + 
Sec.  + 


FIG.  17. 

and  fourth  quadrants,  and  y  is  positive  in  the  first  and 
second.  Hence 

For  an  angle  in  the  first  quadrant  all  functions  are  positive, 
since  x,  y,  r  are  positive. 

In  the  second  quadrant  x  alone  is  negative,  so  that  those 
functions  whose  ratios  involve  x,  viz.  cosine,  tangent,  co- 
tangent, secant,  are  negative;  the  others,  sine  and  cosecant, 
are  positive. 

In  the  third  quadrant  x  and  ^  are  both  negative,  so  that 
those  functions  involving  r,  viz.  sine,  cosine,  secant,  cosecant, 
are  negative  ;  the  others,  tangent  and  cotangent,  are  positiue. 

In  the  fourth  quadrant  y  is  negative,  so  that  sme,  tangent, 
cotangent,  cosecant  are  negative,  and  cosine  and  secant,  positive. 

Values.  In  the  triangle  of  reference  of  any  angle,  the 
hypotenuse  r  is  never  less  than  #  or  ?/.  Then  if  r  be  taken  of 
any  fixed  length,  as  the  angle  varies,  the  base  and  perpen- 
dicular of  the  triangle  of  reference  may  each  vary  in  length 

from  0  to  r.     Hence  the  ratios  -  and  -  can  never  be  greater 

r  r  r  r 

than  1,  nor  if  x  and  y  are  negative,  less  than  —1;  and  — >  - 

x  y 


20 


PLANE   TRIGONOMETRY. 


cannot  have  values  between  +  1  and  —  1.      But  the  ratios 

7/  /7* 

2-  and  —  may  vary  without  limit,  i.e.   from    +00    to  —  oo. 
x  y 

Therefore  the  possible  values  of  the  functions  of  an  angle 
are : 

sine  and  cosine  between  +  1  and  —  1, 

i.e.  sine  and  cosine  cannot  be  numerically  greater  than  1; 

tangent  and  cotangent  between  4-  oo  and  —  oo, 
i.e.  tangent  and  cotangent  may  have  any  real  value  ; 
secant  and  cosecant  between  -|-  oo  and  -h  1,  and  —  1  and  —  oo, 

i.e.  secant  and  cosecant  may  have  any  real  values,  except 
values  between  +  1  and  —  1. 

These  limits  are  indicated  in  the  following  figures.     The 
student  should  carefully  verify. 


Sin  90°=  1 
Cos  90=  ±0 
Tan  90°=  ±00 


Sin  270°=  -1 
Cos  270°= +0 
Tan  270°=±oo 


90° 
F 

Sin.      +1 

+  1 

Cos.      -  0 

+  0 

Ian.   -co 

+00 

•I'       *        (* 

jj 

i    i 

S5       (§       f5 

55 

<S     £ 

+  0,   -1,    -0 

+  o, 

+  1,  +ov  O9 

-0,   -  1,    40 

O    -o, 

+  1,  -0^360° 

Sin.      -1 

-i 

Cos.      -  0 

+  0 

Tan.     +co 

-00 

F' 

270° 

FIG.  18. 


25.  In  tracing  the  changes  in  the  values  of  the  functions  as 
a  changes  from  0°  to  360°,  consider  the  revolving  line  r  as 
of  fixed  length.  Then  x  and  y  may  have  any  length  between 
0  and  r. 

As  a  increases  through 


Sine. 


At  0°,  sin  a  =  &  =  -  =  0. 

r    r 


y. 


. 

the  first  quadrant,  y  increases  from  0  to  r,  whence  -  increases 
from  0  to  1.     In  passing  to  180°  sin  a  decreases  from  1  to  0, 


VARIATIONS  IN   THE   FUNCTIONS.  21 

since  y  decreases  from  r  to  0.  As  a  passes  through  180°,  y 
changes  sign,  and  in  the  third  quadrant  decreases  to  nega- 
tive /*,  so  that  sin  a  decreases  from  0  to  —  1.  In  the  fourth 
quadrant  y  increases  from  negative  r  to  0,  and  hence  sin  a. 
increases  from  —  1  to  0. 

Cosine  depends  on  changing  values  of  x.  Show  that, 
as  a  increases  from  0°  to  360°,  cos  a  varies  in  the  four 
quadrants  as  follows:  1  to  0,  0  to  —  1,  —  1  to  0,  0  to  1. 

Tangent  depends  on  changing  values  of  both  y  and  x. 

At    0°,  y  =  0,  x  =  r,  at  180°,  y  =  0,  x  =  -  r, 

at  90°,  x  =  0,  y  =  r,  at  270°,  x=Q,y  =  —  r. 

Hence  tan  0°  =  -^  =  -  =  0.     As  a  passes  to  90°,  y  increases 
x      r 

to  r,  and  x  decreases  to  0,  so  that  tan  «  increases  from  0  to  oo. 
As  a,  passes  through  90°,  x  changes  sign,  so  that  tan  a 
changes  from  positive  to  negative  by  passing  through  oo. 
In  the  second  quadrant  x  decreases  to  negative  r,  y  to  0,  and 
tan  a  passes  from  —  oo  to  0.  As  a  passes  through  180°, 
tana  changes  from  minus  to  plus  by  passing  through  0, 
because  at  180°  y  changes  to  minus.  In  the  third  quadrant 
tana  passes  from  0  to  oo,  changing  sign  at  270°  by  passing 
through  GO,  because  at  270°  x  changes  to  plus.  In  the  fourth 
quadrant  tan  a  passes  from  —  oo  to  0. 

Cotangent.  In  like  manner  show  that  cot  a  passes  through 
the  values  oo  to  0,  0  to  —  oo,  oo  to  0,  0  to  —  oo,  as  a  passes 
from  0°  to  360°. 

Secant  depends  on  x  for  its  value.  Noting  the  change 
in  x  as  under  cosine,  we  see  that  secant  passes  from  1  to  oo, 
-  oo  to  —  1,  —  1  to  —  oo,  oo  to  1. 

Cosecant  passes  through  the  values  oo  to  1,  1  to  oo, 
—  oo  to  —  1,  —  1  to  —  oo. 

The  student  should  trace  the  changes  in  each  function 
fully,  as  has  been  done  for  sine  and  tangent,  giving  the 
reasons  at  each  step. 


22 


PLANE   TRIGONOMETRY. 


a 

0°  to  90° 

90°  to  180° 

180°  to  270° 

270°  to  360° 

sin 

0  to  1 

1  to  0 

-  0  to  -  1 

-  1  to  -  0 

cos 

1  to  0 

-  0  to  -  1 

-1  to  -0 

0  to  1 

tan 

0  to  oo 

-  oo  to  -  0 

0  to  oo 

-  oo  to  -  0 

cot 

oo  to  0 

-  0  to  -oo 

oo  to  0 

-  0  to  -  co 

sec 

1  to  oo 

—  oo  to  —  1 

—  1  to  —  oo 

oo  to  1 

CSC 

GO    tO    1 

1  to  co 

-  oo  to  -  1 

-  1  to  -oo 

*  26.  Graphic  representation  of  functions.  These  variations 
are  clearly  brought  out  by  graphic  representations  of  the 
functions.  Two  cases  will  be  considered  :  I,  when  a  is  a 
constant  angle  ;  II,  when  a  is  a  variable  angle. 

I.    When  a  is  a  constant  angle. 

The  trigonometric  functions  are  ratios,  pure  numbers. 
By  so  choosing  the  triangle  of  reference  that  the  denomi- 
nator of  the  ratio  is  a  side  of  unit  length,  the  side  forming 
the  numerator  of  that  ratio  will  be  a  geometrical  representa- 
tion of  the  value  of  that  function,  e.g.  if  in  Fig.  19  r  =  1, 

then  sin  «  =  ^  =  ^=y.     This  may  be  done  by  making  a  a 
r      1 

central  angle  in  a  circle  of  radius  1,  and  drawing  triangles 
of  reference  as  follows : 


E 


FIG.  19. 


GRAPHIC   REPRESENTATION  OF   FUNCTIONS.          23 
In  all  the  figures  A  OP  =  a,  and 


.EC1 


OP      OD      OD 

=      == 


OP      OO      OO     nn 

=      =- 


It  appears  then  that,  by  taking  a  radius  1, 

sine  is  represented  by  the  perpendicular  to  the  initial  line, 
drawn  from  that  line  to  the  terminus  of  the  arc  sub- 
tending the  given  angle; 

cosine  is  represented  by  the  line  from  the  vertex  of  the 
angle  to  the  foot  of  the  sine  ; 

tangent  is  represented  by  the  geometrical  tangent  drawn 
from  the  origin  of  the  arc  to  the  terminal  line,  produced 
if  necessary; 

cotangent  is  represented  by  the  geometrical  tangent  drawn 
from  a  point  90°  from  the  origin  of  the  arc  to  the 
terminal  line,  produced  if  necessary  ; 

secant  is  represented  by  the  terminal  line,  or  the  terminal 
line  produced,  from  the  origin  to  its  intersection  with 
the  tangent  line; 

cosecant  is  represented  by  the  terminal  line,  or  the  terminal 
line  produced,  from  the  origin  to  its  intersection  with 
the  cotangent  line. 


24 


PLANE  TRIGONOMETRY. 


These  lines  are  not  the  functions,  but  in  triangles  drawn 
as  explained  their  lengths  are  equal  to  the  numerical  values 
of  the  functions,  and  in  this  sense  the  lines  may  be  said  to 
represent  the  functions.  It  will  be  noticed  also  that  their 
directions  indicate  the  signs  of  the  functions.  Let  the 
student  by  means  of  these  representations  verify  the  results 
of  Arts.  24  and  25. 

II.    When  a  is  a  variable  angle. 

Take  XX'  and  YY'  as  axes  of  reference,  and  let  angle 
units  be  measured  along  the  #-axis,  and  values  of  the  func- 
tions parallel  to  the  #-axis,  as  in  Art.  3.  We  may  write 
corresponding  values  of  the  angle  and  the  functions  thus  : 

a  =  0°,  30°,  45°,    60°,  90°,  120°,  135°,  150°,  180°,  210°,     225°, 
sin«=0,    J,    |V2,.£V3,  1,    JV3,  £V2,    £,       0,      -  Jt  -  J  V2, 

a=  240°,  270°,  300°,  315°,  330°,  360°,  -30°,  -45°,  -60°,  -90°,  etc., 
sin<*=-£\/3,  -1,  -£\/3,  -i\/2,  -£,  0,  -£,  -£\/2,  -|V3,  -1,  etc. 


These  values  will  be  sufficient  to  determine  the  form  of  the 
curve  representing  the  function.     By  taking  angles  between 

those  above,  and  computing 
the  values  of  the  function,  as 
given  in  mathematical  tables, 
the  form  of  the  curve  can  be 
determined  to  any  required 
degree  of  accuracy.  Reduc- 
ing the  above  fractions  to 
decimals,  it  will  be  convenient 
to  make  the  «/-units  large  in 

comparison  with  the  #-units. 
In  the  figure  one  x-uuit  repre- 
sents  15°,  and  one  ?/-unit  0.  25. 
Measuring  the  angle  values  along  the  #-axis,  and  from  these 
points  of  division  measuring  the  corresponding  values  of  sin  a 
parallel  to  the  #-axis,  as  in  Art.  3,  we  have,  approximately, 


Curves  of  Sine  and  Cosecant. 

Cosecant  ^TZI 
FIG.  20. 


GRAPHIC   REPRESENTATION   OF  FUNCTIONS. 


25 


OX1  =  30°  =  2  units,  OX2  =  45°    =3  units, 

Xl  Y1  =  J      =2  units,        ^2  F2  =  0.  71  =  2.  84  units, 


=  60°    =4  units,  etc., 
XBY9  =  0.86  =  3.44  units,  etc. 

We  have  now  only  to  draw  through  the  points  Yv  I"2,  Y3. 
etc.,  thus  determined,  a  continuous  curve,  and  we  have  the 
sine-curve  or  sinusoid. 

The  dotted  curve  in  the  figure  is  the  cosecant  curve.  Let 
the  student  compute  values,  as  above,  and  draw  the  curve. 

In  like  manner  draw  the  cosine  and  secant  curves,  as 
follows  : 


Curves  of  Cosine  and  Secant. 

Cosine 

Secant ~ 

FIG.  21. 


Tangent  curve.     Compute  values  for  the  angle  a  and  for 
tan  «,  as  before  : 

«  =  0°,  30°,  45°,  60°,  90°,  120°,  135°,  150°,  180°,  210°,  225°,  240°,  270°, 
tan  a  =  0,  £\/3,  1,    V3,  ±00,  -  V3,  -1,  -|V3,  0,    ^V3,    1,     V3,  ±00, 


a  =  -  30°,  -  45°,  -  60°,  -  90°,  etc., 
tan  a  =  -  }  V3,  -  1,    -  V3,    ±  o>,  etc. 

Then  lay  off  the  values  of  a  and  of  tan  a  along  the  a?,  and 
parallel  to  the  #-axis,  respectively.     It  will  be  noted  that, 


26 


PLANE  TRIGONOMETRY. 


as  a  approaches  90°,  tan  «  increases  to  oo,  and  when  a  passes 
90°,  tan  a  is  negative.    Hence  the  value  is  measured  parallel 


\ 


Curves  of  Tangent  and  Cotangent. 

Tangent    

Cotangent 


to  the  #-axis  downward,  thus  giving  a  discontinuous  curve, 
as  in  the  figure. 

*  27.  The  following  principles  are  illustrated  by  the  curves  : 

1.  The  sine  and  cosine  are  continuous  for  varying  values 
of  the  angle,  and  lie  within  the  limits  -f  1  and  —  1.     Sine 
changes  sign  as  the  angle  passes  through  180°,   360°,  •••, 
n  180°,  while  cosine  changes  sign  as  the  angle  passes  through 
90°,  270°,  •••,  (2ra  +  l)  90°.     Tangent   and   cotangent  are 
discontinuous,  the  one  as  the  angle  approaches  90°,  270°,  •  ••, 
(2  rc  +  1)  90°,  the  other  as  the  angle  approaches  180°,  360°,  •••, 
w!80°,  and  each  changes  sign  as  the  angle  passes  through 
these  values.     The  limiting  values  of  tangent  and  cotangent 
are  +  oo  and  —  oo. 

2.  A  line  parallel  to  the  ?/-axis  cuts  any  of  the  curves  in 
but  one  point,  showing  that  for  any  value  of  a  there  is  but 
one  value  of  any  function  of  a.     But  a  line  parallel  to  the 
cc-axis  cuts  any  of  the  curves  in  an  indefinite  number  of 
points,  if  at  all,  showing  that  for  any  value  of  the  function 
there  are  an  indefinite  number  of  values,  if  any,  of  «. 


GRAPHIC   REPRESENTATION  OF  FUNCTIONS.  27 

3.  The  curves  afford  an  excellent  illustration  of  the  varia- 
tions in  sign  and  value  of  the  functions,  as  a  varies  from  0 
to  360°,  as  discussed  in  Art.  25.     Let  the  student  trace  these 
changes. 

4.  From  the  curves  it  is  evident  that  the  functions  are 
periodic,  i.e.  each  increase  of  the  angle  through  360°  in  the 
case  of  the  sine  and  cosine,  or  through  180°  in  the  case  of 
the  tangent  and  cotangent,  produces  a  portion  of  the  curve 
like  that  produced  by  the  first  variation  of  the  angle  within 
those  limits. 

5.  The  difference  in  rapidity  of  change  of  the  functions 
at  different  values  of  a  is  important,  and  reference  will  be 
made  to  this  in  computations  of  triangles.     (See  Art.  64, 
Case  III.)     A  glance  at  the  curves  shows  that  sine  is  chang- 
ing in  value  rapidly  at  0°,  180°,  etc.,  while  near  90°,  270°, 
etc.,  the  rate  of  change  is  slow.     But  cosine  has  a  slow  rate 
of  change  at  0°,  180°,  etc.,  and  a  rapid  rate  at  90°,  270°,  etc. 
Tangent  and  cotangent  change  rapidly  throughout. 

Ex.    Let  the  student  discuss  secant  and  cosecant  curves. 

ORAL  WORK. 

1.  Express  in  radians  180°,  120°,  45°;    in   degrees,  Jf  radians,  27r, 
ITT,  ITT. 

2.  If  £  of  a  right  angle  be  the  unit,  what  is  the  measure  of  £  of  a 
right  angle?  of  90°?  of  135°? 

3.  Which  is  greater,  cos  30°  or  £  cos  60°?  tan  -  or  cot-?  sin  -  or  cos-  ? 

634  4 

4.  Express  sin  a  in  terms  of  sec  a ;  of  tan  a ;  tan  a  in  terms  of  cos  a ; 
of  sec  a. 

5.  Given  sin  a  =  f ,  find  tan  a.    If  tan  a  =  1,  find  sin  a,  esc  a,  cot  a ; 
also  tan  2  a,  sin  2  a,  cos  2  a. 

6.  If  cos  a  =  $,  find  sin  -,  tan  -. 

7.  In  what  quadrant  is  angle  t,  if  both  sin  t  and  cos  t  are  minus  ?  if 
sin  t  is  plus  and  cos  t  minus  ?  if  tan  t  and  cot  t  are  both  minus  ?   if  sin  t 
and  esc  t  are  of  the  same  sign  ?     Why  ? 

8.  Of  the  numbers  3,  $,  —  5,  —  |,  a,  —  b,  oo,  0,  which  may  be  a  value 
of  sin  p  ?  of  sec  jo  ?  of  tan  p  ?     Why  ? 


28  PLANE  TRIGONOMETRY. 


EXAMPLES. 

1.   If  sin  26°  40'  =  0.44880,  find,  correct  to  0.00001,  the  cosine  and 
tangent.  % 


2.  If  tan  a  =  \/3,  and  cot  ft  =  |V3,  find  sin  a  cos  ft  -  cos  a  sin  ft. 

J-.  7.   (00  30 

3.  Evaluate  sin  3Q°  cot  3Q°  -  cos  60°  tan  60° 

sin  90°  cos  0° 

Prove  the  identities : 

4.  tan 4(1  -  cot2 4)  + cot  4(1  -  tan24)  =  0. 

5.  (sin  4  4-  sec  4)2  +  (cos  4  +  esc  4)2  =  (1  +  sec  4  esc  4)2. 

6.  sin2  x  cos  x  esc  x  —  cos3  x  esc  x  sin2  x  +  cos4  z  sec  x  sin  #  =  sin3  x  cos  a: 
+  cos3  x  sin  a;. 

7.  tan2  w  -f  cot2  MJ  =  sec2  w  esc2  w  —  2. 

8.  sec2  v  +  cos2  w  =  2  +  tan2  w  sin2  w. 

9.  cos2 1  -f  1  =  2  cos3 1  sec  *  +  sin2 1. 

10.  esc2  f  -  sec2 1  =  cos2 1  csc*-t  -  sin2  *  sec2 1. 

11.  The  sine  of  an  angle  is  —   ~  n  ;  find  the  other  functions. 

m2  +  n2 

12.  If  tan  4  +  sin  4  =  m,  tan  4  —  sin  4  =  w,  prove  m2  —  n2  =  4  Vwm. 
_ —  «* 

Solve  for  one  function  of  the  angle  involvedAthe  equations : 

13.  sin0  +  2cos0  =  1.  16.    2  sin2  a:  +  cos  a:  -  1  =  0. 
14    cos  ft  _  3                                           17-   sec2 a:  -  7  tan  x  -  9  =  0. 

'  tan«~2*  18.   3  cscy  +  lOcoty  -  35  =  O.Cffy 

15.    V3csc20  =  4cot0.  19.   sin2v  -fcosv-  1  =  0. 

20.  a  sec2  w  -f  b  tan  w  +  c  —  a  =  0. 

. 

21.  If  ^B-d  =  V2,  2^5:  =  V3,  find  4  and  5. 

sin  B  tan  jB 

22.  Find  to  five  decimal  places  the  arc  which  subtends  the  angle  of 
1°  at  the  centre  of  a  circle  whose  radius  is  4000  miles.  -, 

23.  If  esc  4  =  |V3,  find  the  other  functions,  when  4  lies  between 
-  and  TT. 

24.  In  each  of  two  triangles  the  angles  are  in  G.  P.     The  least  angle 
of  one  of  them  is  three  times  the  least  angle  of  the  other,  and  the  sum  of 
the  greatest  angles  is  240°.     Find  the  circular  measure  of  each  of  the 
angles.  ^          _    _ 

i^oLV  +  ^^-.H,     *-_  L__;_l_        , 


CHAPTER   III. 

FUNCTIONS   OF   ANY   ANGLE  —  IN  VERSE  FUNCTIONS. 

28.  By  an  examination  of  the  figure  of  Art.  24  it  is  seen 
that  all  the  fundamental  relations  between  the  functions  hold 
true  for  any  value  of  a.     The  table  of  Art.  16  expresses  the 
functions  of  a,  whatever  be  its  magnitude,  in  terms  of  each 
of  the  other  functions  of  that  angle  if  the  ±  sign  be  prefixed 
to  the  radicals. 

The  definitions  of  the  trigonometric  functions  (Art.  12) 
apply  to  angles  of  any  size  anjl  sign,  but  it  is  always  possible 
to  express  the  functions  of  any  angle  in  terms  of  the  func- 
tions of  a  positive  acute  angle. 

The  functions  of  any  angle  0,  greater  than  360°,  are  the 
same  as  those  of  6  ±  n  •  360°,  since  6  and  6  ±  n  •  360°  have 
the  same  triangle  of  reference.  Thus  the  functions  of  390°, 
or  of  750°,  are  the  same  as  the  functions  of  390°-  360°,  or 
of  750°—  2-360°,  i.e.  of  30°,  as  is  at  once  seen  by  drawing  a 
figure.  So  also  the  functions  of  —  315°,  or  of  —  675°  are 
the  same  as  those  of  -  315°  +  360°,  or  of  -  675°  +  2-360°, 
i.e.  of  45°. 

For  functions  of  angles  less  than  360°  the  relations  of  this 
chapter  are  important. 

29.  To  find  the  relations  of  the  functions  of  -  0,  90°  ±  9, 
180°  ±  0,  and  270°  ±  0  to  the  functions  of  6,  6  being  any  angle. 

Four  sets  of  figures  are  drawn,  I  for  0  an  acute  angle,  II 
for  0  obtuse,  III  for  6  an  angle  of  the  third  quadrant,  and 
IV  for  6  an  angle  of  the  fourth  quadrant. 

In  every  case  generate  the  angles  forming  the  compound 
angles  separately,  i.e.  turn  the  revolving  line  first  through 


30 


PLANE   TRIGONOMETRY. 


II 


II 


IV 


FUNCTIONS  OF   ANY  ANGLE. 


31 


(d) 

90^0 


II 


in 


FIG.  23. 


32  PLANE   TRIGONOMETRY. 

0°,  90°,  180°,  or  270°,  and  then  from  this  position  through 
0,  or  —  0,  as  the  case  may  be.  Form  the  triangles  of  refer- 
ence for  (a)  the  angle  6,  (5)  -  0,  (<?)  180°  ±  6,  (d)  90°  ±  0, 
0)  270°  ±0. 

The  triangles  of  reference  (a),  (6),  (V),  (d),  and  (e),  in 
each  of  the  four  sets  of  figures,  I,  II,  III,  IV,  are  similar, 
being  mutually  equiangular,  since  all  have  a  right  angle  and 
one  acute  angle  equal  each  to  each.  Hence  the  sides  x,  y,  r 
of  the  triangles  (a)  are  homologous  to  x',  y',  r'  of  the  cor- 
responding triangles  (5)  and  (<?),  but  to  y',  x',  r'  ,  of  the 
corresponding  triangles  (d)  and  (e).  For  the  sides  x  of 
triangle  (a)  and  xr  of  the  triangles  (£)  and  (<?)  are  opposite 
equal  angles,  and  hence  are  homologous,  but  the  sides  y'  are 
opposite  this  same  angle  in  triangles  (c?)  and  (e),  and  there- 
fore sides  y'  of  (d)  and  (e)  are  homologous  to  x  of  (a). 

Attending  to  the  signs  of  x  and  x',  y  and  y1  in  the  similar 
triangles  (a)  and  (5), 

sin(-0)=^  =  -^  =  -sin0, 

x'      x 

cos(—  #)=  —  =  -      =  cos  0, 
r'      r 

tan  (-  0)  =  ^=  -  ^  =  -  tan0. 

' 


x  x 

Also  in  the  similar  triangles  (a)  and  (<?), 

sin  (180°  -  0)  =  ^  =  y-     =  sin  0, 
cos  (180°  -  0)  =  p  =  -  ?  =  -  cos  0, 
tan  (180°  -  0)  =  ^  =  -  ^  =  -  tan  0. 

3?  3/ 

In  like  manner  show  that 

sin  (180°  +  0)  =  -  sin  0, 
cos(180°  +  0)=-cos0, 
tan  (180°  +  0)=  tan  0. 


FUNCTIONS  OF  ANY  ANGLE. 
Again,  in  the  similar  triangles  (a)  and  (cT), 

sin  (90°  +  (9)  =  ^  =  |     =cos<9, 
cos  (90°  +  0)  =  ^  =  -  y-  =  -  sin  0, 


tan  (90°  +  0)=  ^  =  -  -=  -  cot  0. 

x'          y 

Show  that 

sin  (90°  -  0)  =  cos  0, 

cos  (90°  -0)=  sin  0, 

tan  (90° -0)  =  cot  0. 
Finally,  from  the  similar  triangles  (a)  and  (e),  show  that 

sin  (270°±0)=-cos0, 

cos  (270°±0)=±sin0, 

tan  (270°  ±  0)  =  T  cot  0. 

From  the  reciprocal  relations  the  student  can  at  once- 
write  the  corresponding  relations  for  secant,  cosecant,  and 
cotangent. 

30.  Since  in  each  of  the  four  cases  x',  y'  of  triangles 
(£>)  and  (c)  are  homologous  to  #,  y  of  triangle  (a),  while 
x',  y'  of  the  triangles  (d)  and  (e)  are  homologous  to  «/,  x 
of  triangle  (a),  we  may  express  the  relations  of  the  last 
article  thus  : 

{I      >3 
180°  +  0  corresPond  to  the  same  functions 

of  0,  while  those  of  \  ^^0      «  correspond  to  the  co-function® 
of  0,  due  attention  being  paid  to  the  signs. 

The  student  can  readily  determine  the  sign  in  any  given 
case,  whether  0  be  acute  or  obtuse,  by  considering  in  what 
quadrant  the  compound  angle,  90°  ±  0,  180°  ±  0,  etc.,  would 


34  PLANE   TRIGONOMETRY. 

lie  if  6  were  an  acute  angle,  and  prefixing  to  the  correspond- 
ing functions  of  6  the  signs  of  the  respective  functions  for 
an  angle  in  that  quadrant.  Thus  90°  +  0,  if  6  be  acute,  is 
an  angle  of  the  second  quadrant,  so  that  sine  and  cosecant 
are  plus,  the  other  functions  minus.  It  will  be  seen  that 
sin  (90°  +  6)=  +cos0,  cos  (90°  +  0)=  —  sin0,  etc.,  and  this 
will  be  true  whatever  be  the  magnitude  of  0.  It  will  assist 
in  fixing  in  the  memory  these  important  relations  to  notice 
that  when  in  the  compound  angle  0  is  measured  from  the 
#-axis,  as  in  90°  ±  0,  270°  ±  0,  the  functions  of  one  angle 
correspond  to  the  co-functions  of  the  other,  but  when  in  the 
compound  angle  0  is  measured  from  the  ic-axis,  as  in  ±  #, 
180°  ±  0,  then  the  functions  of  one  angle  correspond  to  the 
same  functions  of  the  other. 

These  relations,  as  has  been  noted  in  Art.  28,  can  be 
extended  to  angles  greater  than  360°,  and  it  may  be  stated 
generally  that 

function  0  =  ±  function  (2  w  •  90°  ±  0), 
function  0  =  ±  co-function  [(2  rc  +  1)  90°  ±  0]. 

Computation  tables  contain  angles  less  than  90°  only.  The  chief 
utility  of  the  above  relations  will  be  the  reduction  of  functions  of  angles 
greater  than  90°  to  functions  of  acute  angles.  Thus,  to  find  tan  130°  20', 
look  in  the  tables  for  cot  40°  20',  or  for  tan  49°  40'.  Why? 

Ex.  1.  What  angles  less  than  360°  have  the  same  numerical  cosine 
as  20°? 

cos  20°  =  -  cos  (180°  ±  20°)  =  cos  (360°  -  20°). 

.-.  200°,  160°,  340°  have  the  same  cosine  numerically  as  20°. 

2.   Find  the  functions  of  135°;  of  210°. 

sin  135°  =  sin  (90°  +  45°)  =  cos  45°  =  £  V2, 

cos  135°  =  cos  (180°  -  45°)  =  -  cos  45°  =  -  i\/2,  etc. 

sin  210°  =  sin  (180°  -f  30°)  =  -  sin  30°  =  -  $. 

Let  the  student  give  the  other  functions  for  each  angle. 


INVERSE  FUNCTIONS.  35 

ORAL  WORK. 

1.  Determine  the  sine  and  tangent  of  each  of  the  following  angles : 
30°,  120°,  -  30°,  -  60°,  f  TT,  2f  TT,  -  135°,  -  TT. 

2.  Which  is  the  greater,  sin  30°  or  sin(-  30°)  ?  tan  135°  or  tan  45°? 
cos  60°  or  cos(  -  60°)  ?  sin  22°  30'  or  cos  67°  30'  ? 

3.  What  positive  angle  has  the  same  tangent  as  — ?    the  same  sine 
as  50°? 

4.  If  tan  6  =  -  1,  find  sin  0. 

5.  Find  sin  510°,  cos(-  60°),  tan  150°. 

6.  Reduce  in  two  ways  to  functions  of  a  positive  acute  angle,  cos  122° 
tan  140°  30',  sin  (-60°). 

7.  Find  all  positive  values  of  x,  less  than  360°,  satisfying  the  fol- 
lowing equations :    cos  x  =  cos  45°,  sin  2  x  =  sin  10°,   tan  3  x  =  tan  60°, 
.sin  x  =  sin  30°,  tan  x  =  tan  135°. 

8.  What  angles  are  determined  when  (a)  sine  and  cosine  are  +  ? 
(ft)  cotangent  and  sine  are  —  ?    (c)  sine  +  and  cosine  —  ?   (d)  cosine  — 

and  cotangent  -f  ?  ^ 

0  ko 

INVERSE   FUNCTIONS. 

31.  That  a  is  the  sine  of  an  angle  6  may  be  expressed  in 
two  ways,  viz.,  sin  0  =  #,  or,  inversely,  0  =  sin"1  a,  the  latter 
being  read,  0  equals  an  angle  whose  sine  is  a,  or,  more  briefly, 
6  is  the  anti-sine  of  a. 

The  notation  sin"1  a,  cos"1  a,  tan"1  a,  etc.,  is  not  a  fortunate  one,  but 
is  so  generally  accepted  that  a  change  is  not  probable.  The  symbol  may 
have  been  suggested  from  the  fact  that  if  ax  =  6,  then  x  =  a~l  Z>,  whence, 
by  analogy,  if  sin  0  =  a,  0  =  sin"1  a.  But  the  likeness  is  an  analogy  only, 
for  there  is  no  similarity  in  meaning.  Sin"1  a  is  an  angle  0,  where  sin  6  =  a, 

and  is  entirely  different  from  (sin  a)-1  = .     In  Europe  the  symbols      rN^ 

sin  a 
arc  sin  a,  arc  cos  a,  etc.,  are  employed. 

32.  Principal  value.     We  have  found  that  in  sin  6  =  a, 
for   any  value   of   0,  a   can   have   but   one  value ;    but  in 
6  =  sin"1  a,  for  any  value  of  a  there  are  an  indefinite  number 
of  values  of  0  (Art.  27,  2). 

Thus,  when  sin  0  =  a,  if  a  =  l,  6  may  be  30°,  150°,  390°, 
510°,  -  330°,  etc.,  or,  in  general,  mr  +(-  1)W30°. 

In  the  solution  of  problems  involving  inverse  functions, 


36 


PLANE   TRIGONOMETRY. 


the  numerically  least  of  these  angles,  called  the  principal 
value,  is  always  used ;  i.e.  we  understand  that  sin"1  a,  tan"1  a, 
are  angles  between  +  90°  and  —  90°,  while  the  limits  of 

a  are  0°  and  180°. 
Thus,     sin-1  J  =  30°,    sin^C  -  })  =  -  30°,    cos"1  J  =  60°, 
cos-1(-i)=120°. 

ORAL  WORK. 

How  many  degrees  in  each  of  the  following  angles?    How  many 
radians? 

1.  cos-^?  »•  ta"-lVS? 

2.  tan-1!? 

3.  cot-^-VS)? 

4.  sin-^ 

5.  cos-H 


6. 


Find  the  values  of  the  functions : 

13.  sin  (tan-1  £\/3). 

14.  tangos-1!). 

15.  tan  (cot-1  [-  GO]). 

16.  cos  (tan"1  oo  ). 

17.  sin  (sin-1  ^V2). 

18.  tan  (tan-1  a;). 


FIG.  24. 


19.  cosCsin-iQ). 

20.  sin  (cos-1  [-  !]). 

21.  cosCcot^VS). 

22.  tan  (sin-1  [-  !]). 

23.  sin  (tan-1  [-  !]). 

Ex.  1.   Construct  cot-1  f . 

Construct  the  right  triangle  xyr,  so  that  x  =  4, 
y  =  3,  whence  jungle  xr  =  cot"1  f . 

2.   Find  cos(tan-1  T85). 
Let  6  =  tan"1  T8^,  whence 

tan  0  =  TV  and  cos  6  =  |f. 
.•.  cos  0  —  cos  (tan"1  -j8^)  =  jf. 


3.  If  0  =  esc-1  a,  prove  6  =  cos"1 


esc     =  a ; 


sin  t^  =  -> 
a 


and 


EXAMPLES.  37 


EXAMPLES. 

1.  Construct  sin'1!,  tan'1^,  cos'^-  |). 

2.  Find  tannin"1  T\),  sin  (tan-1  ^). 

3.  If  0  =  sin-1  a,  prove  0  =  tan-1  —  a 

Vl-a2 

4.  Show  that  sin'1  a  =  90°  -  cos"1  a. 


5.  Prove  tan-1  V3  +  cot^VS  =  -• 

6.  Prove  tan-1  (sin  -\  =  cos-H\/2. 

7.  What  angles,  less  than  360°,  have  the  same  tangent  numerically 
as  I0°? 

8.  Given  tan  143°  22'  =  -  0.74357  ;  find,  correct  to  0.00001,  sine  and 
cosine. 

9.  If  cot2(90°  +  /3)  +  csc(90°  -  ft)  -  1  =  0,  find  tan  J3. 

10.  Find  all  positive  values  of  x,  less  than  360°,  when  sin  x  =  sin  22°  30'  ; 
when  tan  2  x  =  tan  60°. 

11.  When  is  sin  x  =  a  possible,  and  when  impossible  ?     ^  -c  ( 

2ab 

12.  Verify  sin'1  1  +  cos'1—  +  tan-1  \/3  =  sin'1  —  • 

13.  What  values  of  x  will  satisfy  sin-^.r2  -  x)=  30°? 

14.  If  tan2  0  -  sec2  a  =  1,  prove  sec  6  +  tan8  0  esc  0  =  (3  +  tan2  «)i 

15.  Prove  sin  A  (1  +  tan  A)  +  cos  A  (1  +  cot  A)  =  sec  A  +  esc  A. 

16.  Solve  the  simultaneous  equations  : 

sin-1(2  x  +  3  y}  =  30°  and  3  x  +  2  y  =  2. 


17.   Verify  (a)     tan  60°  =  J1  ~  cos  12 
x  1  +  cos  12 

(6)     cos  60°  = 


-  cos  120° 
120°' 

1  -tan2  30° 
1  +  tan2  30°' 

(c)     2  sin2  60°  =  1  -  cos  120°. 


18.   Show  that  the  cosine  of  the  complement  of  -  equals  the  sine  of 

the  supplement  of  -• 
6 


38  PLANE  TRIGONOMETRY. 

REVIEW. 

Before  leaving  a  problem  the  student  should  review  and  master  all 
principles  involved. 

1.  Construct  cos"1 3*7 ;  sin-1(— f);  tan"1 2. 

2.  Find  cos  (sin-1  f ) ;   tan  (cos'1  [-$]). 

3.  Prove  cot'1  a  =  cos"1       q 

VI  -ha2 

4.  Given  a  =  cot-1  f,  find  tan  a  +  sin  (90°  -f-  a). 

5.  Find  tan  ( sin'1!  -f  cos-1—)- 

6.  State  the  fundamental  relations  between  the  trigonometric  func- 
tions in  terms  of  the  inverse  functions.     Thus, 

sin-1  a  =  esc-1-,    sin"1  a  =  cos~1Vl  —  a2,  etc. 

7.  Find  all  the  angles,  less  than  360°,  whose  cosine  equals  sin  120°. 

8.  Given  cot"1 2.8449,  find  the  sine  and  cosine  of  the  angle,  correct 
to  0.0001. 

9.  If  tan2  (180°  -  0)  -  sec  (180°  +  0)  =  5,  find  cos  0. 
10.   If  sin  0  =  1,  find 


11.  Is  sin  x  —  2  cos  z  +  3sina;  —  6  =  0a  possible  equation  ? 

12.  Verify  (a)     sin  60°  =  .,  2  tan  3?0°   • 

1  +  tan2 30° 

(6)     2  cos2  60°  =  1  +  cos  120°. 

(c)     cos  60°  -  cos  90°  =  2  cos2  30°  -  2  cos2  45°. 

13.  If  sin  x  =      *(q  +  2ft)       find  sec  x  and  tan  a?. 

a2  +  2  a&  +  2  62 

14.  Prove   1  +  sin  0  -  cos0  +  1  +  sing  +  cos0  =  g^ 

1  +  sin  0  +  cos  0      1  +  sin  0  —  cos  & 

15.  Prove 

cos  45°  -f  cos  135°  +  cos  30°  +  cos  150°  -  cos  210°  +  cos  270°  =  sin  60°. 

16.  If  tan  0  =  —^ ,  prove  that 

sin  0(1  +  tan 0)  +  cos  6(1  +  cot 6)  -  sec  0  =  | 

17.  Solve  sin2*  -f  sin2 (*  +  90°)  +  sin2 (x  +  180°)  =  1. 


EXAMPLES. 


18.   Given  cos2  ex,  =  m  sin  a  —  n,  find  sin  a. 


20.  Given  tan  238°  =  1.6,  find  sin  148°. 

21.  Prove  tan"1  m  +  cot'1  m  =  90°.  k^UUQ  [ 

22.  Find  sin  (sin"1^  +  cos"1^). 

23.  Solve  cot2  6  (2  esc  0  -  3)  +  3  (esc  0  -  1)  =  0. 

24.  Prove  sin2  a  sec2  /3  +  tan2  ft  cos2  a  =  sin2  a  +  tan2  J3. 

25.  Prove  cos6  F  +  sin6  V  =  1  -  3  sin2  V  +  3  sin4  V. 

26.  What  values  of  A  satisfy  sin  2  A  =  cos  3  A  ? 


27.   If  tan  C  =  —  -  —  ,  and  tan  i>  ss--  -     _^  find  tan  D  in  termg 


28.  If  sin  x  —  cos  x  +  4  cos2  a:  =  2,  find  tan  x  ;  sec  x. 

29.  Does  the  value  of  sec  a:,  derived  from  sec2  a;  =     ~  ~  c0^  x,  give  a 
possible  value  of  x  ? 

30.  Prove 

[cot  (90°  -  A)-  tan  (90°  +  4)]  [sin  (180°  -  A)  sin  (90°  +  ^1)]  =  1. 

31.  Prove  (1  +  sin  Ay  [cot  A  +2  sec  .4(1  -csc^)]  +  csc^4  cos3  A  =  0. 

32.  Given  sin  x  =  m  sin  y,  and  tan  x  =  n  tan  y,  find  cos  z  and  cos  y. 

33.  Given  cot  201°  =  2.6,  find  cos  111°. 

34.  Find  the  value  of 

cos-H  +  sin-HV2  +  csc-^-  1)+  tan^l  -  2cot-1V3. 

35.  Solve  2  cos2  $  +  11  sin  0  -  7  =  0. 

36.  Prove 

cos2  B  +  cos2  (B  +  90°)  +  cos2  (B  +  180°)  +  cos2  (B  +  270°)  =  2. 


CHAPTER   IV. 

COMPUTATION   TABLES. 

33.  Natural  functions.      It  has  been  noted  that  the  trigo- 
nometric functions  of  angles  are  numbers,  but  the  values 
were  found  for  only  a  few  angles,  viz.  0°,  30°,  45°,  60°, 
90°,  etc.     In  computations,  however,  it  is  necessary  to  know 
the  values  of  the  functions  of  any  angle,  and  tables  have 
been  prepared  giving  the  numerical  values  of  the  functions 
of   all   angles   between   0°  and   90°  to   every  minute.     In 
these   tables  the   functions  of   any  given   angle,  and   con- 
versely the  angle  corresponding  to  any  given  function,  can 
be  found  to  any  required  degree  of  accuracy ;  e.g.  by  look- 
ing   in    the   tables  we   find   sin  24°  26'=  0.41363,  and   also 
1 .6415  =  tan  58°  39'.     These  numbers  are  called  the  natural 
functions,  as  distinguished  from  their  logarithms,  which  are 
called  the  logarithmic  functions  of  the  angles. 

Ex.  1.   Find  from  the  tables  of  natural  functions  : 

sin  35°  14';     cos  54°  46';     tan  78°  29';     cos  112°  58';     sin  135°. 

2.    Find  the  angles  less  than  180°  corresponding  to :  * 

sin-1 0.37865;  cos'1 0.37865;   tan-*  0.58670 ;  cos"1 0.00291 ;   siiT*  0.99999. 

34.  Logarithms.     The    arithmetical    processes    of    multi- 
plication,   division,   involution,   and   evolution,   are   greatly 
abridged   by  the  use  of   tables   of   logarithms  of   numbers 
and  of  the  trigonometric  ratios,  which  are  numbers.     The 
principles  involved  are  illustrated  in  the  following  table : 

Write  in  parallel  columns  a  geometrical  progression  having 
the  ratio  2,  and  an  arithmetical  progression  having  the  dif- 
ference 1,  as  follows  : 

40 


LOGARITHMS. 


41 


G.  P. 


1 
2 
4 

8 

16 

32 

64 

128 

256 

512 

1024 

2048 

4096 

8192 

16384 

32768 

65536 

131072 

262144 

524288 


A.  P. 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 


It  will  be  perceived  that  the  numbers  in 
the  second  column  are  the  indices  of  the 
powers  of  2  producing  the  corresponding 
numbers  in  the  first  column,  thus  :  26  =  64, 
211  =  2048,  218  =  262144,  etc.  The  use  of 
such  a  table  will  be  illustrated  by  examples. 

Ex.  1.   Multiply  8192  by  128. 

From  the  table,  8192  =  213,  128  =  2*.  Then  by 
actual  multiplication,  8192  x  128  =  1048576,  or  by  the 
law  of  indices,  213  x  27  =  220  =  1048576  (from  table). 

Notice  that  the  simple  operation  of  addition  is  sub- 
stituted for  multiplication  by  adding  the  numbers  in 
the  second  column  opposite  the  given  factors  in  the 
first  column.  This  sum  corresponds  to  the  number 
in  the  first  column  which  is  the  required  product. 

2.   Divide  16384  by  512. 

16384  •*•  512  =  32,  which  corresponds  to  the  result 
obtained  by  use  of  the  table,  or  214  +  29  =  25  =  32. 
The  operation  of  subtraction  takes  the  place  of 
division. 


3.   Find 


1048576 

In  the  table,  262144  is  opposite  18.  18  -4-  6  =  3, 
which  is  opposite  8,  the  required  root ;  i.e.  simple  division  takes  the 
place  of  the  tedious  process  of  evolution. 


4.  Cube  64. 

5.  Multiply  256  by  4096. 


7.   Divide  1048576  by  32768. 


35.  The  above  table  can  be  made  as  complete  as  desired 
by  continually  inserting  between  successive  numbers  in  the 
first  column  the  geometrical  mean,  and  between  the  opposite 
numbers  in  the  second,  the  arithmetical  mean,  but  in  prac- 
tice logarithms  are  computed  by  other  methods.  The  num- 
bers in  the  second  column  are  called  the  logarithms  of  the 
numbers  opposite  in  the  first  column.  2  is  called  the  base  of 
this  system,  so  that  the  logarithm  of  a  number  is  the  exponent 
~by  which  the  base  is  affected  to  produce  the  number. 


42  PLANE   TRIGONOMETRY. 

Thus,  the  logarithm  of  512  to  the  base  2  is  9,  since 
29  =  512. 

Logarithms  were  invented  by  a  Scotchman,  John  Napier,  early  in  the 
seventeenth  century,  but  his  method  of  constructing  tables  was  different 
from  the  above.  See  Encyc.  Brit.,  art.  "  Logarithms,"  for  an  exceedingly 
interesting  account.  De  Morgan  says  that  by  the  aid  of  logarithms  the 
labor  of  computing  has  been  reduced  for  the  mathematician  to  about 
one-tenth  part  of  the  previous  expense  of  time  and  labor,  while  Laplace 
has  said  that  John  Napier,  by  the  invention  of  logarithms,  lengthened 
the  life  of  the  astronomer  by  one-half. 

Columns  similar  to  those  above  might  be  formed  with  any 
other  number  as  base.     For  practical  purposes,  however,  10 
is  always  taken  as  the  base  of  the  system,  called  the  common 
system,  in  distinction  from  the  natural  system,  of  which  the 
n  base  is  2.71828  •••,  the  value  of  the  exponential  series  (Higher 
1    Algebra).     The  natural  system  is  used  in  theoretical  discus- 
sions.     It  follows  that  common  logarithm's  are  indices,  positive 
or  negative,  of  the  powers  of  10. 

Thus,  103  =  1000  ;  i.e.  log  1000  =  3  ; 

10-2  =  ^=0.01;  i.e.  log 0.01  =  -2. 

36.  Characteristic  and  mantissa.     Clearly  most  numbers 
are  not  integral  powers  of  10.     Thus  300  is  more  than  the 
second  and  less  than  the  third  power  of  10,  so  that 

log  300  =  2  plus  a  decimal. 

Evidently  the  logarithms  of  numbers  generally  consist  of 
an  integral  and  a  decimal  part,  called  respectively  the  charac- 
teristic and  the  mantissa  of  the  logarithms. 

37.  Characteristic  law.      The  characteristic  of  the  loga- 
rithm of  a  number  is  independent  of  the  digits  composing 
the    number,   but   depends  on   the  position  of  the  decimal 
point,  and  is  found  by  counting  the  number  of  places  the  first^ 
significant  figure  in  the  number   is  removed  from  the   units'" 
place,  being  positive  or  negative  according  as  the  first  significant 


LOGARITHMS.  43 

figure  is  at  the  left  or  the  right  of  units'  place.  This  follows 
from  the  fact  that  common  logarithms  are  indices  of  powers 
of  10,  and  that  10re,  n  being  a  positive  integer,  contains  n  +  1 
places,  while  10~w  contains  n  —  1  zeros  at  the  right  of  units' 
place.  Thus  in  146.043  the  first  significant  figure  is  two 
places  at  the  left  of  units'  place  ;  the  characteristic  of  log 
146.043  is  therefore  2.  In  0.00379  the  first  significant  digit 
is  three  places  at  the  right  of  units'  place,  and  the  charac- 
teristic of  log  0.00379  is  -  3. 

To  avoid  the  use  of  negative  characteristics,  such  charac- 
teristics are  increased  by  10,  and  — 10  is  written  after  the 
logarithm.  Thus,  instead  of  log  0.00811  =  3.90902,  write 
7.90902  —  10.  In  practice  the  —  10  is  generally  not  written, 
but  it  must  always  be  remembered  and  accounted  for  in  the 
result. 

Ex.   Determine  the  characteristic  of  the  logarithm  of : 

1;  46;  0.009;  14796.4;  230.001;  105  x  76;  0.525;  1.03;  0.000426. 
O      I         7-,o          f  6  f/O      O          £-/• 

38.  Mantissa  law.  The  zfrantissa  of  the  logarithm  of  a 
number  is  independent  of  the  position  of  the  decimal  point, 
but  depends  on  the  digits  composing  the  number,  is  always 
positive,  and  is  found  in  the  tables. 

For,  moving  the  decimal  point  multiplies  or  divides  a 
number  by  an  integral  power  of  10,  i.e.  adds  to  or  subtracts 
from  the  logarithm  an  integer,  and  hence  does  not  affect  the 
mantissa.  Thus, 

log   225. 67  =  log  225. 67, 

log   2256. 7  =  log 225. 67  xlO1    =  log 225.67  +  1, 
log 22567.0  =  log 225.67  x  102    =  log 225.67  +  2, 
log   22.567  =  log 225.67  x  10'1  =  log 225.67  +(-  1), 
log  0. 22567  =  log  225. 67  x  10~3  =  log  225. 67  +  ( -  3), 

so  that  the  mantissse  of  the  logarithms  of  all  numbers  com- 
posed of  the  digits  22567  in  that  order  are  the  same,  .35347. 
Moving  the  decimal  point  affects  the  characteristic  only. 
The  student  must  remember  that  the  mantissa  is  always  positive. 


44  PLANE  TRIGONOMETRY. 

Log  0.0022567  is  never  written  -  3  +.35347,  but  3.35347,  the  minus 
sign  being  written  above  to  indicate  that  the  characteristic  alone  is  nega- 
tive. In  computations  negative  characteristics  are  avoided  by  adding 
and  subtracting  10,  as  has  been  explained. 

39.  We  may  now  define  the  logarithm  of  a  number  as  the 
index  of  the  power  to  which  a  fixed  number,  called  the  base, 
must  be  raised  to  produce  the  given  number. 

Thus,  ax  —  6,  and  x  =  logab  (where  loga&  is  read  logarithm 
of  b  to  the  base  a)  are  equivalent  expressions.  The  relation 
between  base,  logarithm,  and  number  is  always 

(base)log  =  number. 

To  illustrate :  Iog28  =  3  is  the  same  as  23  =  8;  Iog381  =  4  and 
34=  81  are  equivalent  expressions  ;  and  so  are  Iog101000  =  3 
and  103  =  1000,  and  Iog100.001=  -3  and  10-3  =  0.001. 

Find  the  value  of  : 

log, 64;  Iog5125;  Iog3243  ;  loga(a)*;  Iog27  3  ;  logxl. 

40.  From  the  definition  it  follows  that  the  laws  of  indices 
apply  to  logarithms,  and  we  have : 

I.  The  logarithm  of  a  product  equals  the  sum  of  the  loga- 
rithms of  the  factors. 

II.  The  logarithm  of  a  quotient  equals  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor. 

III.  The  logarithm  of  a  power  equals   the   index   of  the 
power  times  the  logarithm  of  the  number. 

IV.  The  logarithm  of  a  root  equals  the  logarithm  of  the 
number  divided  by  the  index  of  the  root. 

For  if  ax  =  n  and  ay  =  m, 

then  nxm  =  ax+y,    .-.  lognm  =  x  +  y  =•  logn  +  logm; 

and  n  +  m  =  ax-y,    /.log—   =  x  —  y  =  logn  —  logra; 

m 

also  nr=  (axy=  arx,     .:  log  nr   =  rx  =  r  x  log  n  ; 

finally,     </n  =  Vo*  =  ar,      .'.  log^/n  =  -  =  -  log  n. 


LOGARITHMS.  45 

EXAMPLES. 
Given  log  2  =  0.30103,  log  3  =  0.47712,  log  5  =  0.69897,  find : 

1.  log 4.  4.   log 9.  7.   Iogl53.  10.   logV|f 

2.  log  6.  5.   log  25.  8.   logf.  '    .p-j-gg 

3.  log  10.  6.  logV3.  9-   log  15  x  9. -11-   ^V^OO 

USE  OF  TABLES. 
41.  To  find  the  logarithm  of  a  number. 

First.    Find  the  characteristic,  as  in  Art.  37. 

Second.    Find  the  mantissa  in  the  tables,  thus  : 

(a)  When  the  number  consists  of  not  more  than  four 
figures. 

In  the  column  N  of  the  tables  find  the  first  three  figures, 
and  in  the  row  N  the  fourth  figure  of  the  number.  The 
mantissa  of  the  logarithm  will  be  found  in  the  row  opposite 
the  first  three  figures  and  in  the  column  of  the  fourth  figure. 

Illustration.    Find  log 42.38. 

The  characteristic  is  1.     (Why  ?) 

In  the  table  in  column  N  find  the  figures  423,  and  on  the 
same  page  in  row  N  the  figure  8.  The  last  three  figures  of 
the  mantissa,  716,  lie  at  the  intersection  of  column  8  and 
row  423.  To  make  the  tables  more  compact  the  first  two 
figures  of  the  mantissa,  62,  are  printed  in  column  0  only. 
Then  log 42.38  =  1.62716. 

Find  log  0. 8734  =  1. 94121, 

log  3.5       =  log  3.500  =  0.54407, 
log  36350  =4.56050. 

(6)  When  the  number  consists  of  more  than  four  figures. 

Find  the  mantissa  of  the  logarithm  of  the  number  com- 
posed of  the  first  four  figures  as  above.  To  correct  for  the 
remaining  figures  we  interpolate  by  means  of  the  principle  of 
proportional  parts,  according  to  which  it  is  assumed  that,  for 
differences  small  as  compared  with  the  numbers,  the  differences 


46  PLANE  TRIGONOMETRY. 

between  several  numbers  are  proportional  to  the  differences  be* 
tween  their  logarithms. 

The  theorem  is  only  approximately  correct,  but  its  use 
leads  to  results  accurate  enough  for  ordinary  computations. 

Ex.  1.   To  find  log  89.4562. 

As  above,  mantissa  of  log  894500  =  0.95158, 
mantissa  of  log  894600  =  0.95163, 

.-.  log  894600  -  log  894500  =  0.00005,  called  the  tabular  difference. 
Let  log  894562  -  log  894500  =  x  hundred-thousandths. 

Now,  by  the  principle  of  proportional  parts, 

log  894562  -  log  894500  =  894562  -  894500 
log  894600  -  log  894500      894600  -  894500' 

or         -  =  — ,  whence  x  =  .62  of  5  =  3.1 
o      100 

.-.  log  89.4562  =  1.95158  +  0.00003  =  1.95161, 

all  figures  after  the  fifth  place  being  rejected  in  five-place  tables.  If, 
however,  the  sixth  place  be  5  or  more,  it  is  the  practice  to  add  1  to  the 
figure  in  the  fifth  place.  Thus,  if  x  =  0.0000456,  we  should  call  it 
0.00005,  and  add  5  to  the  mantissa. 

~~2.   Find  log  537.0643. 

To  interpolate  we  have  x  :  9  =  643  :  1000,  i.e.  x  =  5.787; 

.-.  log  537.0643  =  2.72997  +  0.00006. 
~3.   Find  log  0.0168342  =  2.22619. 

4.   Find  log  39642.7  =  4.59816. 

42.   To  find  the  number  corresponding  to  a  given  logarithm. 

The  characteristic  of  the  logarithm  determines  the  posi- 
tion of  the  decimal  point  (Art.  37). 

(a)  If  the  mantissa  is  in  the  tables,  the  required  number 
is  found  at  once. 

Ex.  1.  Find  log-1 1.94621  (read,  the  number  whose  logarithm  is 
1.94621). 

The  mantissa  is  found  in  the  tables  at  the  intersection  of  row  883  and 

column  5. 

.-.  log-1 1.94621  =  88.35, 

the  characteristic  1  showing  that  there  are  two  integral  places. 


LOGARITHMS.  47 

(5)  If  the  exact  mantissa  of  the  given  logarithm  is  not  in 
the  tables,  the  first  four  figures  of  the  corresponding  num- 
ber are  found,  and  to  these  are  annexed  figures  found  by 
interpolating  by  means  of  the  principle  of  proportional 
parts,  as  follows : 

Find  the  two  successive  mantissse  between  which  the  given 
mantissa  lies.  Then,  by  the  principle  of  proportional  parts, 
the  amount  to  be  added  to  the  four  figures  already  found  is 
such  a  part  of  1  as  the  difference  between  the  successive 
mantissse  is  of  the  difference  between  the  smaller  of  them 
and  the  given  mantissa. 

2.    Find  log-1  1.43764. 

Mantissa  of  log  2740  =  0.43775 

of  log  2739  =  0.43759 
Differences  1  16 

Mantissa  of  log  required  number  =  0.43764 

of  log  2739  =  0.43759 

Differences  x  5 

By  p.  p.  x  :  1  =  5  :  16  and  x  =  -£$  =  0.3125. 

Annexing  these  figures,  log-1  1.43764  =  27.3931+. 

"    3.    Find  log'1 1.48762. 

The  differences  in  logarithms  are  14,  6. 

,*=A  =  .428+, 

and  log-1  1.48762  =  0.307343+. 

4.   Find  log  891.59;  log  0.023  ;  log|;  log  0.1867;  log  \/2. 
-    5.   Find  log"1  2.21042  ;  log'1  0.55115;  log-1 1.89003. 

43.  Logarithms  of  trigonometric  functions.  These  might 
be  found  by  first  taking  from  the  tables  the  natural  func- 
tions of  the  given  angle,  and  then  the  logarithms  of  these 
numbers.  It  is  more  expeditious,  however,  to  use  tables 
showing  directly  the  logarithms  of  the  functions  of  angles 
less  than  90°  to  every  minute.  Functions  of  angles  greater 
than  90°  are  reduced  to  functions  of  angles  less  than  90°  by 


48  PLANE   TRIGONOMETRY. 

the  formulae  of  Art.  29.  To  make  the  work  correct  for 
seconds,  or  any  fractional  part  of  a  minute,  interpolation 
is  necessary  by  the  principle  of  proportional  parts,  thus  : 

Ex.1.   Find  log  sin  28°  32' 21". 

In  the  table  of  logarithms  of  trigonometric  functions,  find  28°  at  the 
top  of  the  page,  and  in  the  minute  column  at  the  left  find  32'.  Then 
under  log  sin  column  find  log  sin  28°  32'  =  9.67913  -  10 

log  sin  28°  33  =  9.67936  -  10 
Differences         1'  23 

By  p.  p.          x  :  23  =  21"  :  60",  i.e.  x  =  —  x  23  =  8.05. 

.-.  log  sin  28°  32' 21"  =  9.67913  +  0.00008'-  10 
=  9.67921  -  10. 

Whenever  functions  of  angles  are  less  than  unity,  i.e.  are  decimals 
(as  sine  and  cosine  always  are,  except  when  equal  to  unity,  and  as  tan- 
gent is  for  angles  less  than  45°),  the  characteristic  of  the  logarithm  will 
be  negative,  and,  accordingly,  10  is  always  added  in  the  tables,  and  it 
must  be  remembered  that  10  is  to  be  subtracted.  Thus,  in  the  example 
above,  the  characteristic  of  the  logarithm  is  not  9,  but  1,  and  the  log- 
arithm is  not  9.67913,  as  written  in  the  tables,  but  9.67913  -  10. 

2.   Find  log  cos  67°  27' 50". 

In  the  table  of  logarithms  at  the  foot  of  the  page,  find  67°,  and  in  the 
minute  column  at  the  right,  27'.  Then  computing  the  difference  as 
above,  x  =  25. 

But  it  must  be  noted  that  cosine  decreases  as  the  angle  increases 
toward  90°.  Hence,  log  cos  67°  27'  50"  is  less  than  log  cos  67°  27',  i.e. 
the  difference  25  must  be  subtracted,  so  that 

log  cos  67°  27'  50"  =  9.58375  -  0.00025  -  10 
=  9.58350  -  10. 

44.  To  find  the  angle  when  the  logarithm  is  given,  find  the 
successive  logarithms  between  which  the  given  logarithm 
lies,  compute  by  the  principle  of  proportional  parts  the 
seconds,  and  add  them  to  the  less  of  the  two  angles  corre- 
sponding to  the  successive  logarithms.  This  will  not  neces- 
sarily be  the  angle  corresponding  to  the  less  of  the  two 
logarithms ;  for,  as  has  been  seen,  the  number,  and,  theref ore, 
the  logarithm,  may  decrease  as  the  angle  increases. 


LOGARITHMS.  49 

Ex.  1.  Find  the  angle  whose  log  tan  is  9.88091. 

log  tan  37°  14'  =  9.88079  -  10 
log  tan  37°  15'  =  9.88105  -  10 

Differences       60"  26 

log  tan  37°  14'  =  9.88079  -  10 
log  tan  angle  required  =  9.88091  -  10 

Differences  x"  12 

.-.  x  :  60  =  12  : 26,    or    x"  =  £f  x  60"  =  28",   approximately,  and  the 
angle  is  37°  14'  28". 

2.  Find  the  angle  whose  log  cos  =  9.82348. 

We  find  x  =  T6¥  x  60"  =  26",  and  the  angle  is  48°  14'  26". 

3.  Show  that        log  cos    25°  31'  20"  =  9.95541 ; 

log  sin  110°  25'  20"  =  9.97181 ; 
log  tan   49°  52'  10"  =  0.07417. 

4.  Show  that  the  angle  whose  log  tan  is  9.92501  is  40°  4'  39"  ;  whose 
log  sin  is  9.88365  is  49°  54'  18" ;  whose  log  cos  is  9.50828  is  71°  11'  49". 

45.   Cologarithms.     In  examples  involving  multiplications 
and  divisions  it  is  more  convenient,  if  n  is  any  divisor,  to 

add  log  -  than  to  subtract  log  n.     The  logarithm  of  :-  is 
called  the  cologarithm  of  n.     Since 

log  -  =  log  1  —  log  n  —  0  —  log  n, 

it  follows  that  colog n  =  —  log  %,  i.e.  logn  subtracted  from 
zero.     To  avoid  negative  results,  add  and  subtract  10. 

Ex.  1.  Find  colog  2963. 

log  1  =  10.00000  -  10 
log  2963=    3.47173 

.-.  colog  2963  =    6.52827  -  10 

2.  Find  colog  tan  16°  17'. 

log  1  =  10.00000  -  10 
log  tan  16°  17'  =    9.46554  -  10 

.•.  colog  tan  16°  17'  =    0.53446 


50  PLANE  TRIGONOMETRY. 

By  means  of  the  definitions  of  the  trigonometric  functions,  the  parts 
of  a  right  triangle  may  be  computed  if  any  two  parts,  one  of  them  being 
a  side,  are  given.    Thus, 

given  a  and  A  in  the  rt.  triangle  ARC. 

Then      c  =  a  -H  sin  A,  b  =  a  -r-  tan  A, 
and  £  =  90° -.4. 


—  j  -  i,          Again,  if  a  and  b  are  given,  then 

FIG.  25.  tan  4  =  -,  c  =  a  -s-  sin  A,  and  5  =  90°  -  A 

b 

3.  Given  c  -  25.643,  B  =  37°  25'  20",  compute  the  other  parts. 

A  =  90°  -  37°  25'  20"  =  52°  34'  40". 
a  =  c  cos  B.  b  =  a  tan  B. 

log  c  =  1.40897  log  a  =  1.30889 

log  cos  B  =  9.89992  log  tan  B  =  9.88376 

log  a  ='1.30889  log  b  =  1.19265 

.-.  a  =  20.365.  .-.  b  =  15.583. 

CAecfc:  c2  =  a2  +  62  =  20.3652  +  15.5832  =  657.57  =  2S.6432. 

4.  Given  b  =  0.356,  5  =  63°  28'  40",  compute  the  other  parts. 

A  =  26°  31'  20". 


sin  5  tan  .6 

log  &  =  9.55145  log  6  =  9.55145 

colog  sin  B  =  0.04829  colog  tan  B  =  9.69816 

log  c  =  9.59974  log  a  =  9.24961 

c  =  0.3979  a  =  0.1777 

Check:  c2  -  a2  =  0.1583  -  0.03157  =  0.12673  =  b2. 

EXAMPLES. 

Compute  the  other  parts  : 

1.  Given  a  =  9.325,  A  =  43°  22'  35". 

2.  Given    c  =  240.32,  a  =  174.6. 

3.  Given  JB  =  76°  14'  23",  a  =  147.53. 

4.  Given  a  =  2789.42,  b  =  4632.19. 

5.  Given  c  =  0.0213,  A  =  23°  14". 

6.  Given  b  =  2,  c  =  3. 


CHAPTER  V. 

APPLICATIONS. 

46.  Many  problems  in  measurements  of  heights  and  dis- 
tances may  be  solved  by  applying  the  preceding  principles. 
By  means  of  instruments  certain  distances  and  angles  may 
be  measured,  and  from  the  data  thus  determined  other 
distances  and  angles  computed.  The  most  common  instru- 
ments are  the  chain,  the  transit,  and  the  compass. 

The  chain  is  used  to  measure  distances.  Two  kinds  are  in 
use,  the  engineer's  chain  and  the  G-unter**  chain.  They  each 
contain  100  links,  each  link  in  the  engineer's  chain  being 
12  inches  long,  and  in  the  Gunter's  7.92  inches. 


FIG.  26. 

The  transit  is  the  instrument  most  used  to  measure  hori- 
zontal angles,  and  with  certain  attachments  to  measure  verti- 
cal angles.  The  figure  shows  the  form  of  the  instrument. 

61 


52 


PLANE   TRIGONOMETRY. 


The  mariners  compass  is  used  to  determine  the  directions, 
or  bearings,  of  objects  at  sea.  Each  quadrant  is  divided 
into  8  parts,  making  the  32  points  of  the  compass,  so  that 
each  point  contains  11°  15'. 


*&$•£ 


w-» 


V*  4V 

"*  *^^ 


epression 


FIG.  28. 


*> 


FIG.  27. 

47.  The  angle  between  the  horizontal  plane  and  the  line 
of  vision  from  the  eye  to  the  object  is  called  the  angle  of 

elevation,  or  of  depression,  according 
as  the  object  is  above  or  below  the 

observer. 

It  is  evident  that  the  elevation 
angle  of  B,  as  seen  from  A,  is  equal 
to  the  depression  angle  of  A,  as  seen  from  B,  so  that  in  the 
solution  of  examples  the  two  angles  are  interchangeable. 

PROBLEMS. 

48.  Some   of   the   more   common   problems   met  with   in 
practice  are  illustrated  by  the  following  : 

To  find  the  height  of  an  object 
when  the  foot  is  accessible. 

The  distance  BO,  and  the  eleva- 
tion angle  B  are  measured,  and  # 
is  determined  from  the  relation 
#  =  BO  tan  B. 


FIG.  29. 


APPLICATIONS. 


53 


Ex.  1.   The  elevation  angle  of  a  cliff  measured  from  a  point  300  ft. 
from  its  base  is  found  to  be  30°.     How  high  is  the  cliff  ? 


Then 


EC  =  300,  B  =  30°. 

x  =  300  •  tan  30°  =  300  -  |V3  =  100\/3. 


2.  From  a  point  175  ft.  from  the  foot  of  a  tree  the  elevation  of  the 
top  is  found  to  be  27°  19'.  Find  the  height  of  the  tree. 

The  problem  may  be  solved  by  the  use  of  natural  functions,  or  of 
logarithms.  The  work  should  be  arranged  for  the  solution  before  the 
tables  are  opened.  Let  the  student  complete. 

BC  =  175.    B  =  27°  19'. 
Then  x  —  EC  tan  B.  Or  by  natural  functions, 

EC  =  175 
tan  B  =  0.5165 


.-.  x  =  90.3875. 


To  find  the  height  of  an  object 
when  the  foot  is  inaccessible. 

Measure  BB',  6  and  6'. 
Then 


. 

cot  6          cot  6 

But  B'  0  =  x  cot  Q\  whence  substituting, 


x 


cot  6  —  cot  0' 

which  is  best  solved  by  the  use  of  the  natural  functions  of 
B  and  0'  '. 

3.  Measured  from  a  certain  point  at  its  base  the  elevation  of  the 
peak  of  a  mountain  is  60°.  At  a  distance  of  one  mile  directly  from  this 
point  the  elevation  is  30°.  Find  the  height  of  the  mountain. 

BB'  =  5280  ft.,    6  =  30°,    &  =  60°. 

y  +  5280 
cot  30° 
5280 


x  = 


cot  30°  -  cot  60' 


But  y  =  x  cot  60°. 
=  4572.48  ft. 


54 


PLANE   TRIGONOMETRY. 


In  surveying  it  is  often  necessary  to  make  measurements 
across  a  stream  or  other  obstacle  too  wide  to  be  spanned  by 
a  single  chain. 

To  find  the  distance  from  C  to  a 
point  B  on  the  opposite  side  of  a 
stream. 

At  C  measure  a  right  angle,  and 
take     CA     a     convenient     distance. 
Measure  angle  A,  then 
FlG-31-  BC  =CA-  tan  A. 

4.  Find  CB  when  angle  A  =  47°  16',  and  CA  =  250  ft. 

5.  From  a  point  due  south  of  a  kite  its   elevation  is  found  to  be 
42°  30';  from  a  point  20  yds.  due  west 

from  this  point  the  elevation  is  36°  24'. 
How  high  is  the  kite  above  the  ground  ? 

A  B  =  x.  cot  42°  30', 
A  C  =  x  .  cot  36°  24', 


.-.  x2-  (cot2  36°  24'  -  cot2  42°  30')  =  400, 
whence 


*>  =  -,  and  x=-  =  24.84  yds. 

.6489  .805  J 


EXAMPLES. 


FIG.  32. 


1.  What  is  the  altitude  of  the  sun  when  a  tree  71.5  ft.  high  casts 
a  shadow  37.75  ft.  long? 

2.  What  is  the  height  of  a  balloon  directly  over  Ann  Arbor  when 
its  elevation  at  Ypsilanti,  8  miles  away,  is  10°  15'? 

3.  The  Washington  monument  is  555  ft.  high.     How  far  apart  are 
two  observers  who,  from  points  due  east,  see  the  top  of  the  monument 
at  elevations  of  23°  20'  and  47°  30',  respectively? 

4.  A  mountain  peak  is  observed  from  the  base  and  top  of  a  tower 
200  ft.  high.     The  elevation  angles  being  25°  30'  and  23°  15',  respec- 
tively, compute  the  height  of  the  mountain  above  the  base  of  the  tower. 

5.  From  a  point  in  the  street  between  two  buildings  the  elevation 
angles  of  the  tops  of  the  buildings  are  30°  and  60°.     On  moving  across 


APPLICATIONS.  55 

the  street  20  ft.  toward  the  first  building  the  elevation  angles  are  found 
to  be  each  45°.  Find  the  width  of  the  street  and  the  height  of  each 
building. 

6.  From  the  peak  of  a  mountain  two  towns  are  observed  due  south. 
The  first  is  seen  at  a  depression  of  48°  40',  and  the  second,  8  miles  farther 
away  and  in  the  same  horizontal  plane,  at  a  depression  of  20°  50'.     What 
is  the  height  of  the  mountain  above  the  plane? 

7.  A  building  145  ft.  long  is  observed  from  a  point  directly  in  front 
of  one  corner.     The  length  of  the  building  subtends  tan-1  3,  and  the 
height  tan-1  2.     Find  the  height. 

8.  An  inaccessible  object  is  observed  to  lie  due  N.E.     After  the  ob- 
server has  moved  S.E.  2  miles,  the  object  lies  N.N.E.     Find  the  distance 
of  the  object  from  each  point  of  observation. 

9.  Assuming  the  earth  to  be  a  sphere  with  a  radius  of  3963  miles, 
find  the  height  of  a  lighthouse  just  visible  from  a  point  15  miles  distant 
at  sea. 

10.  The  angle  of  elevation  of  a  tower  120  ft.  high  due  north  of  an 
observer  was  35° ;  what  will  be  its  angle  of  elevation  from  a  point  due 
west  from  the  first  point  of  observation  250  ft.?    Also  the  distance  of 
the  observer  from  the  base  of  the  tower  in  each  position  ? 

11.  A  railway  5  miles  long  has  a  uniform  grade  of  2°  30' ;  find  the  rise 
per  mile.     What  is  the  grade  when  the  road  rises  70  ft.  in  one  mile  ? 

(The  grade  depends  on  the  tangent  of  the  angle.) 

12.  The  foot  of  a  ladder  is  in  the  street  at  a  point  30  ft.  from  the 
line  of  a  building,  and  just  reaches  a  window  22^  ft.  above  the  ground. 
By  turning  the  ladder  over  it  just  reaches  a  window  36  ft.  above  the 
ground  on  the  other  side  of  the  street.     Find  the  breadth  of  the  street. 

13.  From  a  point  200  ft.  from  the  base  of  the  Forefathers'  monument 
at  Plymouth,  the  base  and  summit  of  the  statue  of  Faith  are  at  an  eleva- 
tion of  12°  40'  48"  and  22°  2' 53",  respectively;   find  the  height  of  the 
statue  and  of  the  pedestal  on  which  it  stands. 

14.  At  a  distance  of  100  ft.  measured  in  a  horizontal  plane  from  the 
foot  of  a  tower,  a  flagstaff  standing  on  the  top  of  the  tower  subtends  an 
angle  of  8°,  while  the  tower  subtends  an  angle  of  42°  20'.      Find  the 
length  of  the  flagstaff. 

15.  The  length  of  a  string  attached  to  a  kite  is  300  ft.     The  kite's- 
elevation  is  56°  6'.     Find  the  height  of  the  kite. 

16.  From  two  rocks  at  sea  level,  50  ft.  apart,  the  top  of  a  cliff  is  ob- 
served in  the  same  vertical  plane  with  the  rocks.     The  angles  of  eleva- 
tion of  the  cliff  from  the  two  rocks  are  24°  40'  and  32°  30'.     What  is  the 
height  of  the  cliff  above  the  sea? 


CHAPTER  VI. 

GENERAL  FORMULAE  —  TRIGONOMETRIC   EQUATIONS 
AND   IDENTITIES. 

49.  Thus  far  functions  of  single  angles  only  have  been 
considered.  Relations  will  now  be  developed  to  express 
functions  of  angles  which  are  sums,  differences,  multiples, 
or  sub-multiples  of  single  angles  in  terms  of  the  functions 
of  the  single  angles  from  which  they  are  formed. 

First  it  will  be  shown  that, 

sin  (a  ±  p)  =  sin  a  cos  p  ±  cos  a  sin  p, 
cos  (a  ±  p)  =  cos  a  cos  p  T  sin  a  sin  p 


1  Ttanatanp 

The  following  cases  must  be  considered  : 

1.  «,  /3,  a  +  /3  acute  angles. 

2.  «,  /3,  acute,  but  a  -f-  ft  an  obtuse  angle. 

3.  Either  a,  or  /3,  or  both,  of  any  magnitude,  positive  or 
negative. 

The  figures  apply  to  cases  1  and  2. 


FIG.  33. 


Let  the  terminal  line  revolve  through  the  angle  «,  and 
then  through  the  angle  /3,  to  the  position  OB,  so  that  angle 

56 


GENERAL   FORMULAE.  57 

• 

XOB  =a  +  f3.  Through  any  point  P  in  OB  draw  perpen- 
diculars to  the  sides  of  «,  DP  and  <7P,  and  through  O  draw 
a  perpendicular  and  a  parallel  to  OX,  MO  and  NO. 

Then  the  angle  QOA  =  a  (why?),  and  (TAT*  is  the  triangle 
of  reference  for  angle  QCP  =  90°  +  «. 

CJVP  is  sometimes  treated  as  the  triangle  of  reference  for  angle  CPN. 
The  fallacy  of  this  appears  when  we  develop  cos  («  +  /?),  in  which  PC 
would  be  treated  as  both  plus  and  minus. 

Now     sin  («  +  /3) 

or  expressing  in  trigonometric  ratios, 

=  MO    OC^.NP     OP 
~(K?'-OP     CP  '  OP 

=  sin  a  cos  /3  -f  sin  (90°  +  a)  sin/3. 

Hence,  since  sin  (90°  -f  a)  =  cos  a,  we  have 
sin  («  +  /3)  =  sin  a.  cos/3  +  cos  a  sin  /3. 
In  like  manner 


.or  expressing  in  trigonometric  ratios, 

ON    CP 


OC     OP      OP    OP 

=  cos  a  cos  /3  +  cos  (90°  +  a)  sin  /3. 
And  since  cos  (90°  +  «)  =  —  sin  a,  we  have 
cos  (a  +  /3)  =  cos  a  cos  /3  —  sin  a  sin  /3. 

It  will  be  noted  that  the  wording  of  the  demonstration  ap- 
plies to  both  figures,  the  only  difference  being  that  when  «  +  /3 
is  obtuse  OD  is  negative.  ON  is  negative  in  each  figure. 

50.  In  the  case,  when  a,  or  /3,  or  both,  are  of  any  magni- 
tude, positive  or  negative,  figures  may  be  constructed  as 
before  described  by  drawing  through  any  point  in  the  terminal 
line  of  ft  a  perpendicular  to  each  side  of  a,  and  through  the  foot 
of  the  perpendicular  on  the  terminal  line  of  a  a  perpendicular 
and  a  parallel  to  the  initial  line  of  a.  Noting  negative  lines, 


58  PLANE   TRIGONOMETRY. 

the  demonstrations  already  given  will  be  found  to  apply  for 
all  values  of  a  and  ft. 

To  make  the  proof  complete  by  this  method  would  require  an  unlim- 
ited number  of  figures,  e.g.  we  might  take  a  obtuse,  both  a  and  ft  obtuse, 
either  or  both  greater  than  180°,  or  than  360°,  or  negative  angles,  etc. 

Instead  of  this,  however,  the  generality  of  the  proposition 
is  more  readily  shown  algebraically,  as  follows  : 

Let  a'  =  90°  +  a  be  any  obtuse  angle,  and  «,  ft,  acute 
angles. 

Then 

sin  (a'  4-  ft)  =  sin  (90°  +  a  +  ft)  =  cos  (a  +  ft) 
=  cos  a  cos  ft  —  sin  a  sin  ft 
=  sin  (90°  4-  a)  cos  ft  +  cos  (90°  +  a)  sin  £(why?) 

=  sin  a'  cos  ft  4-  cos  a!  sin  ft. 
0  ^ 

In  like  manner,  considering  any  obtuse  angle  ft'  =  90°  +  ft, 

it  can  be  shown  that 

sin  («'  +  ft')  =  sin  a'  cos  ft1  4-  cos  a'  sin  ft'  . 
Show  that  cos  («'  +  ftf)  =  cos  a1  cos  £'  -  sin  «'  sin  £'. 

By  further  substitutions,  e.#.  «"  =  90°  ±  «',  /3"  =  90°  ±  0', 
etc.,  it  is  clear  that  the  above  relations  hold  for  all  values, 
positive  or  negative,  of  the  angles  a  and  ft. 

Since  a  and  ft  may  have  any  values,  we  may  put  —  ft  for  ft, 
and  sin  (<*  4-  [—  ft~\) 

=  sin  («  —  £)=  sin  a  cos  (  -  ft)  +  cos  a  sin  (  —  ft) 

=  sin  «  cos  ft  —  cos  a  sin  £  (why  ?). 

Also  cos  (a  —  ft)  =  cos  a  cos  (  —  ft)  —  sin  a  sin  (  —  /3) 

=  cos  a  cos  ft  4-  sin  <*  sin  /3. 
Finally, 

,        ~,  =  sin  (a  ±  ft)  __  sin  «  cos  /3±  cos  «  sin  /3 
*  *  «  " 


cos   a  ±     )      cos  a  cos  ft  T  sin  a  sin 


sin  a  cos  /3     cos  «  sin 


cos  «  cos  ft     cos  a  cos  ft       taiift±tanff 
cos  ft  cos  ft     sin  a  sin  ft      1  qp  tan  a  tan  ft 
cos  a  cos  ft     cos  a  cos  ft 

/ 

'•-  3 


EXAMPLES.  59 

ORAL  WORK. 
By  the  above  formulae  develop  : 

1.  sin  (2  A  +  3  J5)  .  7.  sin  90°  =  sin  (45°  +  45°). 

2.  cos~(90°  -  B).  8.  cos  90°. 

3.  tan  (45°  +  <£).  9.  tan  90°. 

4.  sin  2  A  =  sin  (J.  +  4).  10.  sin  (90°  +  (3  +  y). 

5.  cos  20.  Jll.  cos  (270°  -  m  -  n). 

6.  tan  (180°  +  C).  12.  tan  (90°  +  m  +  n). 

Ex.  1.   Find  sin  75°. 

sin  75°  =  sin  (45°  +  30°)  =  sin  45°  cos  30°  +  cos  45°  sin  30° 


V2      2        V2    2        2V2 
2.   Find  tan  15°. 

tan  15°  =  tan  (45°  -  30°)  =  tan  45°  ~  tan  30° 


1  +  tan  45°  tan  30° 


3.  Prove   sin3yl  - 

sin  A         cos  A 


Combining,  aitlu^  cos^4-cos3J.  sinJ.  _  sin (3  4  -  A) 

sin  A  cos  A  sin  A  cos  A 

sin  2^4      _  sin  (A  +  A)  _  sin  .4  cos  A  +  cos  ^4  sin  A 
sin  ^4  cos  A      sin  J.  cos  .4  sin  A  cos  ^4 

4.  Prove  tan-1  a  +  tan-1  &  =  tan-1  a  +  b  • 

I  —  ab 

Let  a  -  tan-1  a,  /?  =  tan-1  &,  y  =  tan-1  q  +     • 

1  —  ab 

Hence,  tan  a  =  a,  tan  B  =  b,  tan  y  =  a  +     • 

1  —  ab 

Then  a  +  /2  =  y,  and  hence  tan  («  +  /8)  =  tany. 

Expanding,  tan«  +  tan^        a 

1  —  tan  a  tan  /? 

Substituting,  £jt —  =  a  +     » 

1  —  a&      1  —  a& 


60  PLANE   TRIGONOMETRY. 

EXAMPLES. 

1.  Find  cos  15°,  tan  75°. 

2.  Prove  cot  (a  ±  0)  = 


coift±cota 

3.   Prove  geometrically  sin  («  +  ft)  =  sin  a  cos  ft  +  cos  a  sin  $ 
and  cos  («  +  ft)  =  cos  a  cos  ft  —  sin  a  sin  /3, 
given  (a)  ot  acute,  /3  obtuse  ; 

(6)   a,  /?,  obtuse  ; 
(c)   a,  p1,  either,  or  both,  negative  angles. 


t^*.   Prove  geometrically  tan  («  +  £)=  tan  <*  +  tan  /?  . 

1  —  tan  a  tan 


tan  a  tan  ft 

Verify  the  formula  by  assigning  values  to  a  and  ft,  and  finding  the 
values  of  the  functions  from  the  tables  of  natural  tangents. 

5.  Prove  cos  («  +  ft)  cos  (<x  —  ft)  =  cos2  a  —  sin2  p\ 

6.  Show  that  tan  a  +  tan  ft  =  sin  <a  +  ff  . 

cos  or  cos  p 

7.  Given  tan  a  =  ^,  tan  ft  =  f,  find  sin  (a  +  /?) 

8.  Given  sin  280°  =  s,  find  sin  170°. 

9.  If  a  =  67°  22',  0  =  128°  40',  by  use  of  the  tables  of  natural  func- 
tions verify  the  formulae  on  page  56. 

10.   Prove  tan-1 


1  -Vox 
11.   Prove  tan-1 


12.  Prove  sec-1  —  a        =  sin"1  -• 

Va2  -  a;2  « 

13.  If  a  +  ft  =  <o,  prove  cos2  a  +  cos2/?  -  2  cos  «  cos  ftcos<a  =  sin2  o>. 

14.  Solve  i  sin  6  =  1  -  cos  6. 

15.  Prove  sin  (A  +  J5)  cos  ^L  —  cos  (A  +  J5)  sin  A  =  sin  5. 

16.  Prove  cos  (A  +  B)  cos(A  -B)  +  siu(A  +  B)sin(A  -  B)  =  cos2B. 

17.  Prove  sin  (2  a  -  ft)  cos  (a  -  2  ft) 

-  cos  (2  a  -  ft)  sin  (a  -  2  ft)  =  sin  (a  +  ft). 

18.  Prove  sin(n-l)a  cos(n  +  l)a-f  cos(n-l)«sin(n  +  l)a  =  sin  2  na. 

19.  Prove  sin  (135°  -  0)  +  cos  (135°  +  0)  =  0. 


ADDITION— SUBTRACTION   FORMULA.  61 

20.  Prove  1  —  tan2  a  tan2  (3  =  cos  P  ~  sin  <*, 

cos2  a  cos2  p 

21.  Prove  tan<*  +  tanff  =  tan  <*  tan  R 

cot  a  +  cot  p 

22    tan2  (-  -  ot\  =  *  ~  2  S*n  "  COS  **  ' 

\4         /      1  -|-  2  sin  a  cos  a 

51.  The  following  formulae  are  very  important  and  should 
be  carefully  memorized.  They  enable  us  to  change  sums 
and  differences  to  products,  i.e.  to  displace  terms  by  factors. 

sin  6  +  sin  <|>  =  2  sin— —^  cos 

a    i    J. 

sine  -  sin<|>  = 

COS0  +  COS<|>  = 

Q     I       I  Q  i 

cos  0  -  cos<|>  =  -  2  sin  -—-*  sin  — * • 

Since  sin  («  +  yS)  =  sin  a  cos  /3  4-  cos  a  sin  ft . 

and  sin  («  —  /3)  =  sin  a  cos  /S  —  cos  a  sin  ft 

then  sin  (a  +  /3)  +  sin  (a  —  /S)  =  2  sin  a  cos  ft          (1) 

and  sin  (a  +  /3)  —  sin  (a  —  /3)  =  2  cos  a  sin  ft          (2) 

Also  since  cos  («  +  /5)  =  cos  a  cos  /3  —  sin  a  sin  ft 

and  cos  (a  —  /3)=  cos  a  cos  /3  +  sin  a  sin  ft 

then  cos  (a  +  /3)  +  cos  (a  —  /3)  =  2  cos  a  cos  ft         (3) 

and  cos  (a  4-  £)  —  cos  (a  —  /3)  =  —  2  sin  «  sin  ft      (4) 

Put  a  +  yS  =  (9 

and  a  —  (3  =  $ 

2  a  =  #  +  <£,  and  «  = -£, 


Substituting  in  (1),   (2),   (3),  (4),  we  have  the   above 
formulae. 


62  PLANE  TRIGONOMETRY. 

EXAMPLES. 


1.  30 


cos  2  0  +  cos  0  2 

By  formulae  of  last  article  the  first  member  becomes 

2  sin  <^  cos  *! 

2         2_        30 

tt' 


2    Prove  sm  ^  +  2  sm  3  ^  +  sin  5  a  _  sin  3  a 
sin  3  a  -f  2  sin  5  a  +  sin  7  a  ~~  sin  5  a* 

(sin  a  +  sin  5  «)  +  2  sin  3  a   _  2  sin  3  a  cos  2  «  +  2  sin  3  a 
(sin  3  a  +  sin  7  a)  +  2  sin  5  a  ~  2  sin  5a  cos  2  a  +  2  sin  5  a 

_  (cos  2  re  +  1)  sin  3  a  _  sin  3  a 
(cos  2  a  +  1)  sin  5  a     sin  5  a 


3.    Prove  -  -  =  ^  „       „ 

cos  (4  4  -2B)+cos(4B-24) 


cos  (A  + 
4.   Prove  sin  50°  -  sin  70°  +  sin  10°  =  0. 


=  tM 


2  cos  50°  +  70°  sin  50°  ~  70°  =  2  cos  60°  sin  (  -  10°)  =:  -  sin  10°. 


5.   Prove 


sin  4  a  sin  3  a  —  sin  2  a  sin  5  a  +  sin  4  a  sin  7  a 
By  (3)  and  (4),  p.  61, 

cos  5  a  +  cos  a  —  cos  9  «  —  cos  5  a  +  cos  1 1  a  +  cos  9  a 
cos  a  —  cos  7  ot  —  cos  3  a  +  cos  7  a  4-  cos  3  a  —  cos  11  a 

=  cos«  +  cosll«  =  2cos6«cos5«  =  cot6acot5tt> 
cos  a  —  cos  11  a     2  sin  6  a  sin  5  a 

ORAL  WORK. 
By  the  formulae  of  Art.  51  transform : 

6.  cos  5  a  +  cos  a.  8.   2  sin  3  0  cos  0. 

7.  cos  a  —  cos  5  a.  9.   sin  2  a  —  sin  4  a. 


FUNCTIONS  OF  THE  DOUBLE  ANGLE. 


63 


10.  cos  9  6  cos  2  0. 

11.  sin  0  +  sin  -. 

12.  sin  75°  sin  15°. 

13.  cos  7^-0082  p. 

14.  cos  (2,  +  3  ,)  sin  (*,  -  8  ,). 

15.  sin-sin- 


16.  cos  (30°  +  2  <£)  sin  (30°  -  <£). 

17.  sin  (2  r  +  s)  +  sin  (2  r  -  s). 
ia  cos  (2  0  -  a)  -  cos  3  a. 

19.  sin  36°  -H  sin  54°. 

^  ^  6QO  +  ^  ^ 

21  sill  30o  +  cos  30o. 


Prove: 


22.  si"  «  +  si"  ^  =  tan  «-±£eot 

•  sin  a  —  sin  p  2  2 


cos  ^    —  cos  a 

sin  a;  +  sin  y  =  tft 
cos  a:  +  cosy 

sin  a;-  sin  y  =  ^_ 
cos  a;  —  cos  y 


26.   cos  55°  +  sin  25°  =  sin  85°. 
+ 


Simplify:       27.  - 

cos  B  +  cos  *2B  +  cos  3  B 

28    sin  C  -  sin  4  C  +  sin  1C-  sin  10  C 
cos  C  —  cos  4  C  +  cos  7  C  —  cos  10  C 

r-  v      52.  Functions  of  ah  angle  in  terms  of  those  of  the  half  angle. 

If  in  sin  (a  4-  /3)  =  sin  a  cos  /3  +  cos  a  sin  /3,  a  =  /3, 
then       sin  (a  4-  a)  =  sin  2  a  =  2  sin  a  cos  a. 
In  like  manner 

cos  (a  +  a)  =  cos  2  a  =  cos2  a  -  sin3 a 


and 


tan  2  a  = 


2  tan  a 
1  -  tan«  a 


64  PLANE  TRIGONOMETRY. 

ORAL  WORK. 
Ex.     Express  in  terms  of  functions  of  half  the  given  angles : 

1.  sin  4  a.  4.   cos  a:.  6.   sin(2jt?  —  q). 

2.  cos  3  p.  .    a  7.   cos  (30°  +  2  <£). 

5.    sin  ^—  o 

3.  tan  5  t.  8.   sin  (x  +  ?/). 

9.   From  the  functions  of  30°  find  those  of  60° ;  from  the  functions  of 
45°,  those  of  90°. 

53.   Functions  of  an  angle  in  terms  of  those  of  twice  the  angle. 

By  Art.  52,     cos  «  =  1  -  2  sin2  -  =  2  cos2-  -  1. 

2,  2t 

.*.  2  sin2-  =  1  —  cos  a,  and   2  cos2-  =  1  4-  cos  «. 


sm- 

a  2         ^  /I  -  cos  a 


.-.  tan-  = 

2     cos«          M  +  cosa 

Explain  the  significance  of  the  ±  sign  before  the  radicals. 
Express  in  terms  of  the  double  angle  the  functions  of 
120°;  50°;  90°,  with  proper  signs  prefixed. 

Ex.  1.    Express  in  terms  of  functions  of  twice  the  given  angles  each 
of  the  functions  in  Examples  1-8  above. 

2.  From  the  functions  of  45°  find  those  of  22°  30' ;  from  the  functions 
of  36°,  those  of  18°  (see  tables  of  natural  functions). 

3.  Find  the  corresponding  functions  of  twice  and  of  half  each  of  the 
following  angles,  and  verify  results  by  the  tables  of  natural  functions : 

Given  sin  26°  42'  =  0.4493, 

tan  62°  24'  =  1.9128, 
cos  21°  34'  =  0.9300. 

1    #2 


EXAMPLES. 


65 


""  6.   If  A,  B,  C  are  angles  of  a  triangle,  prove 

A  fy*  B        C 
sin  A  +  sin  C  +  sinB  =  4  cos  —  sin  —  sin  —• 

7.  If  cos2 «  +  cos2  2  a  +  cos2  3  a  =  1,  then 

cos  a  cos  2  a  cos  3  a  =  0. 

8.  Prove  cot  A  -  cot  2  A  =  esc  2  4. 


9.  Prove 


1  —  tan 


' 


tana 


l+tan 


2  sin 


tan  («  +  <£)  sin  (2  ct 

11.  If  y  =  tan-1  Vl  +  ^2  +  ^ 

12.  Prove  tan-1  Vl  +  a:'2-  1  _j 

13.  Ify  =  sin-1 


sn 


>1 


tan_1 


1  -a 
,  prove  z  =  tan  y. 


=  5  tan_1 


/I  +  x* 

14.  Prove  cos2  a  +  cos2  /3  —  1  =  cos  («  +  /?)  cos  (a  —  /:?). 

15.  Prove  V(cos  «  —  cos  /?)2  -f  (sin  #  —  sin  /3)2  —  2  sin  — 

16.  Prove  sin-1  A/ — ^—  =  tan-1  -\p  =  -  cos-1  a  ~  x- 

*a  +  x  va      2  a  +  o; 

17.  Prove  cos2  0  —  cos2  <£  =  sin  (^>  +  0)  sin  (<^>  —  0). 

18.  Prove  tan  A  +  tan  (A  +  120°)  +  tan  (^4  -  120°)  =  3  tan  3  A. 

19.  Prove  tan  a  -  tan  ^  =  tan  |?  sec  a. 

rt  Q 

20.  3  tan-1  a  =  tan-1 


l-3a2 

21.  cos2  3  A  (tan2  3  A  -  tan2 .4)  =  8  sin2 A  cos  2  A. 

22.  1  -f  cos  2  (4  -  5)  cos  2  £  =  cosM  +  cos2  (.4  - 

23.  cot2(?  +  ^^gcsc^-sec^ 

2  esc  2  0  +  sec  0 


66  .     PLANE  TRIGONOMETRY. 

TRIGONOMETRIC   EQUATIONS  AND  IDENTITIES. 

54.  Identities.     It  was  shown  in  Chapter  I  that 

sin2  0  +  cos2  0=1 
is  true  for  all  values  of  0,  and  in  Chapter  VI,  that 

sin  (a  4-  ft)  =  sin  a  cos  ft  +  cos  a  sin  ft 
is  true  for  all  values  of  a  and  ft.     It  may  be  shown  that 

sin  2  A 

— —  =  tan  A 

1  +  cos  2  A 

is  true  for  all  values  of  A,  thus : 

sin  2  A      __     2  sin  A  cos  A     (by  trigonometric  transf  orma- 
1  +  cos  2  A      1  +  2  cos2  J.  -  1        tion) 

= -  (by  algebraic  transformation) 

cosJ. 

=  tan  A  (by  trigonometric  transformation). 

Such  expressions  are  called  trigonometric  identities.      They 
are  true  for  all  values  of  the  angles  involved. 

55.  Equations.     The  expression 

2  cos2  «-3cos<*  +  l  =  0 

is  true  for  but  two  values  of  cos  a,  viz.  cos  a=  ^  and  1,  i.e. 
the  expression  is  true  for  a  =  0°,  60°,  300°,  and  for  no  other 
positive  angles  less  than  360°.  Such  expressions  are  called 
trigonometric  equations.  They  are  true  only  for  particular 
values  of  the  angles  involved. 

56.  Method  of  attack.     The  transformations  necessary  at 
any  step  in  the  proof  of  identities,  or  the  solution  of  equa- 
tions,  are   either   trigonometric,   or   algebraic;   i.e.    in  prov- 
ing an  identity,  or  solving  an  equation,  the  student  must 
choose  at  each  step  to  apply  either  some  principles  of  algebra, 
or  some  trigonometric  relations.     If  at  any  step  no  algebraic 
operation  seems  advantageous,  then  usually  the  expression 

4 


METHOD  OF  ATTACK.  67 

should  be  simplified  by  endeavoring  to  state  the  different 
functions  involved  in  terms  of  a  single  function  of  the  angle, 
or  if  there  are  multiple  angles,  to  reduce  all  to  functions  of  a 

single  angle. 

Algebraic 

Transformations  ^ 

Trigonometric,     f  Single  function 

to  change  to  a  1  Single  angle 

No  other  transformations  are  needed,  and  the  student  will 
be  greatly  assisted  by  remembering  that  the  ready  solution 
of  a  trigonometric  problem  consists  in  wisely  choosing  at 
each  step  between  the  possible  algebraic  and  trigonometric 
transformations.  Problems  involving  trigonometric  func- 
tions will  in  general  be  simplified  by  expressing  them  entirely 
in  terms  of  sine  and  cosine. 


EXAMPLES. 

i     P™,  sin  3  A      cos  3  A      9 

1.   Jrrove  — : — —  =  ^. 

sin  A         cos  A 

B    algebra     sin3^  _  cos3^  _  sin  3  A  cos  ^4  -  cos  3  A  sin  A 
sin  J.        cos  ^4  sin  A  cos  ^4 

sin  (3  A  -A)          sin  2  A 

by  trigonometry,  =  — A— -^  =  - — 

sin  A  cos  A        sin  ^4  cos  ^4 

_  2  sin  A  cos  ^4  _  g 

sin  A  cos  -4 
Or,  by  trigonometry, 

sin  3  A      cos  3  A  _  3  sin  A  —  4  sin3  TJ  _  4  cos8  ^4—3  cos  A 
sin  ^4        cos  ^4  sin  A  cos  ^4 

by  algebra,  =3  —  4  sin2  A  —  4  cos2  ^4+3 

=  6  -  4(sin2  A  +  cos2 ^4)  =  2. 
sec  8  B  -  1      tan  8  B 


2.   Prove 


sec  4  0  -  1      tan  2 


No  algebraic  operation  simplifies.  Two  trigonometric  changes  are 
needed.  1.  To  change  the  functions  to  a  single  function,  sine  or  cosine. 
2.  To  change  the  angles  to  a  single  angle,  8^4,  4^4,  or  2^4. 


68  PLANE   TRIGONOMETRY. 

By  trigonometry  and  algebra, 

1  -  cos  8 0     sin  SB 

cos  8 0     _  cos  8  0  ^ 
1  -  cos  4  B  ~  sin  2  0 ' 

cos  4  0         cos  2  0 

by  algebra  cos  4  0(1  -  cos  8  0)  _  sin80cos20 . 

1  -  cos  4  6  sin  2  0 

by  trigonometry, 

cos  40(1-1  +  2  sin2  4  0)  =  2  sin  4  0  cos  4  0  cos  2  0 . 
l-l  +  2sin220  sin20 

by  algebra,  5HL||  =  2  cos  2  0 ; 

and  sin  4  0  =  2  sin  2  0  cos  2  0, 

which  is  a  trigonometric  identity. 

3.  Solve  2  cos2  0  +  3  sin  0  =  0. 

By  trigonometry,  2(1  -  sin2  0)  +  3  sin  0  =  0, 
a  quadratic  equation  in  sin  0. 

By  algebra,  2  sin2  0  -  3  sin  0  -  2  =  0, 

and  (sin  0  -  2)  (2  sin  0  +  1)  =  0. 

.-.  sin  0  =  2,  or  -  |.  Verify. 

The  value  2  must  be  rejected.     Why  ? 

.-.  0  =  210°,  and  330°  are  the  only  positive  values  less  than  360°  that 
satisfy  the  equation. 

4.  Solve  sec  0  -  tan  0  =  2. 

Here  tan  0  =  —  0.75,  .*.  from  the  tables  of  natural  functions, 

0  =  143°  7'  48",  or  323°  7'  48". 
Find  sec  0,  and  verify. 

5.  Solve  2  sin  0  sin  3  0  -  sin2  20  =  0. 

By  trigonometry,      cos  2  0  —  cos  4  0  —  sin2  20  =  0, 
also  cos  20  -  cos220  +  sin2 20  -  sin220  =  0. 

By  algebra,  cos  2  0(1  -  cos  2  0)  =  0. 

.-.  cos  2  0  =  0  or  1, 

and  20  =  90°,  270°,  0°,  or  360°, 

whence  0  =  45°,  135°,  0°,  or  180°.    Verify. 


TRIGONOMETRIC  EQUATIONS.  69 

Or,  by  trigonometry, 

2  sin  0(3  sin  0  -  4  sin3  0)  -  4  sin2  0  cos2  0  =  0; 
by  trigonometry  and  algebra, 

6  sin2  0-8  sin4  0-4  sin2  0  +  4  sin4  0  =  0; 
by  algebra,  2  sin2  0-4  sin4  0  =  0, 

and  2  sin2  0(1  -  2  sin2  0)  =  0. 

.-.  sin  0  =  0,  or  ±  \/|, 
and  0  =  0°,  180°,  45°,  135°,  225°,  or  315°. 

The  last  two  values  do  not  appear  in  the  first  solution,  because  only 
angles  less  than  360°  are  considered,  and  the  solution  there  gave  values 
of  2  0,  which  in  the  last  two  cases  would  be  450°  and  630°. 

Solve  :  C  6.  tan  0  =  cot  0.  8.   2  cos  2  0  -  2  sin  0=1.       _ 

Q    7.  sin20  + pos0  =  1.  9.   sin  2  0  cos  0  =  sin  0. 

Prove :     10.  2  cot  2  A  =  cot  A  -  tan  A . 

11.  cos  2  x  +  cos  2  y  =  2  cos  (x  +  y}  cos  (x  —  y). 

12.  (cos  a.  -\-  sin  #)2  =  1  +  sin  2  a. 

57.   Simultaneous  trigonometric  equations. 

13.    Solve  cos  (x  +  y)  +  cos  (x  -  y}  =  2, 

sin|  +  sin|  =  0. 
By  trigonometry, 

cos  x  cos  ?/  —  sin  x  sin  ?/  +  cos  x  cos  ?/  +  sin  x  sin  y  =  2, 
so  that  cos  x  cos  y  =  1 ; 

also, 

and  .•.  cos  x  =  cos  y. 

Substituting,  cos2  a;  =  1, 

cos  x  =  ±  1. 
.-.  x  =  0°,  or  180°, 
and  y  =  x  =  0°,  or  180%   Verify. 


70  PLANE  TRIGONOMETRY 

14.   Solve  for  R  and  F. 

W  -  Fsin  i  -  R  cos  i  =  0, 

W+  Fcosi  -  Rsini  =  0. 
To  eliminate  F, 

W  cos  i  —  F  sin  i  cos  i  —  R  cos2  i  =  0, 

W  sin  i  +  Fcos  i  sin  i  —  R  sin2  i  =  0. 
Adding,  FP(sin  {  +  cos  i)  -  #(sin2  i  +  cos2  i)  =  0. 

.-.  R  =  W(sin  i  +  cos  i). 
Substituting,    TF  —  Fsin  i  —  W(sin  i  +  cos  i)cos  i  =  0 

.    F  =  W  —  TT(sin  i  +  cos  t)  cos  i 

sin  i 
If  W  =  3  tons,  and  i  =  22°  30',  compute  F  and  72. 

^  =  3(0.3827  +  0.9239)=  3.9198. 

p      3  -  3(0.3827  +  0.9239)0.9239  _  _  -,  fi24 

0.3827 
Solve : 

-~15.  472  cot  6  -  263  cot  <£  =  490,  307  cot  6  -  379  cot  <£  =  0. 

16.  sin  2  #  +  1  =  cos  x  +  2  sin  a: . 

17.  cos2  6  +  sin  0  =  1. 

18.  If  2A(cos20-sin20)-2asin0cos0  +  2&sin0cos0  =  0,   prove 


a-b 
Prove: 

19.  tan  y  =  (1  +  sec  y}  tan  ^« 

20.  2  cot-1  x  =  esc-1  *  *  *2- 

21.  sin(<£  +  45°)  +  sin  (<£  +  135°)  =  V2  cos  <f>. 

22     cos  v  +  cos  3  y 1 t 

cos  3  v  +  cos  5  v      2  cos  2  v  —  sec  2 1? 

23.  cos  3  a:  —  sin  3  x  =  (cos  x  +  sin  #)  (1  —  2  sin  2  z). 
Solve : 

24.  sin  2  0  +  sin  0  =  cos  2  0  -f  cos  0. 

25.  4  cos(0  +  60°)  -  V2  =  \/6  -  4  cos  (0  +  30°). 

26.  cot  2  0  =  tan  0-1. 

27.  cos  0  +  cos  2  0  +  cos  3  0  =  0. 


TRIGONOMETRIC   EQUATIONS.  71 

28.  sin  2  a:  +\/3cos2#  =  1. 

29.  3  tan2j»  +  8  cos2;?  =  7. 

30.  Determine  for  what  relative  values  of  P  and  W  the  following 
equation  is  true: 


31.  Compute  N  from  the  equation  ^V+  —  cos  a  -—sin  a-Wcosa  =  Q, 

3  3 

when  W  =  2000  pounds  and  a  satisfies  the  equation  2  sin  a  =  1  +  cos  a. 

32.  sin  0  -  tan  <£(cos  6  +  sin  0)  =  cos  0,  sin  0  -  tan  <j>  cos  0  =  1. 
Prove:  °  ***',*  S^i' <K   ( 

4cos2*      *' 


^  =  2^-l-zfifig- 
34.   sin-1  f  -  sin-1  T5g  =  sin'1  £f .     =    ^^  ^  _   ^J^L- 

^r\o)~  ~r^3TCfT) 


35     tan  I  —  4-  —  \ A/*  ~t~  atli  <"  -5,1 '*^v  ' 

\4      2  /       *1  —  sin  o> 

36.  2  sin'1  \  —  cos"1 1. 

37.  If  sin  A   is  a  geometric  mean  between  sin  B  and  cos  J5,  prove 
cos  2  ^4  =2  sin(45  -  B) cos  (45  +  B). 

38.  Prove     sin  (a  +  ft  +  y)  =  sin  a  cos  /:?  cos  y  +  cos  a  sin  /?  cos  y 

+  cos  a  cos  ft  sin  y  —  sin  a  sin  j8  sin  y. 
Also  find  cos(a  +  ft  +  y). 

39.  Prove  tan(«  +  B  +  y)  =  tan  "  +  tan  ^  +  tan  7  ~  ta"  f<  ta"  ^  ta'L7. 

1  —  tan  «  tan  ft  —  tan  /?  tan  y  —  tan  y  tan  a 

If  a,  /?,  and  y  are  angles  of  a  triangle,  prove 

40.  tan  a  +  tan  /?  +  tan  y  =  tan  a  tan  /?  tan  y. 

41.  cot  ^  +  cot  "  +  cot  ^  —  cot  -  cot  ^  cot  2- 

If  «  +  ft  +  y  =  90°,  prove 

42.  tan  a  tan  ft  +  tan  /:?  tan  y  +  tan  y  tan  a  =  1. 
Prove : 

43.  sin  na  =  2  sin  (n  —  1)  a  cos  a  —  sin  (n  —  2)«. 

44.  cos  na  =  2  cos  (n  —  1)  «  cos  a  —  cos  (n  —  2)a. 

45.  tann«=   tan(n  -  1)«  +  tan  «  . 

1  —  tan  (n  —  1)  a  tan  a 


CHAPTER  VII. 

TRIANGLES. 

58.  In  geometry  it  has  been  shown  that  a  triangle  is 
determined,  except  in  the  ambiguous  case,  if  there  are  given 
any  three  independent  parts,  as  follows  : 

I.    Two  angles  and  a  side. 
II.    Two  sides  and  an  angle, 
(a)  the  angle  being  included  by  the  given  sides, 
(5)  the  angle  being  opposite  one  of  the  given  sides  (am- 
biguous case). 
III.    Three  sides. 

The  angles  of  a  triangle  are  not  three  independent  parts,  since  they 
are  connected  by  the  relation  A  +  B  +  C  =  180°. 

The  three  angles  of  a  triangle  will  be  designated  A,  B,  0, 
the  sides  opposite,  a,  b,  c. 

But  the  principles  of  geometry  do  not  enable  us  to  compute 
the  unknown  parts.  This  is  accomplished  by  the  following 
laws  of  trigonometry  : 

T      T         _p  „.  sin  A      sinB      sin  0 

1.    Liaw  of  bines, 

a 

II.   Law  of  Tangents,   tan          "        =    ZL     etc. 


III.    Law  of  Cosines,  cos  A  —  -  —  -  —  —  ,  etc. 

2  be 

59.    Law  of  Sines.     In  any  triangle  the  sides  are  propor- 
tional to  the  sines  of  the  angles  opposite. 

Let  ABO  be  any  triangle,  p  the  perpendicular  from  B 
on  b.     In  I  (Fig.  34),  0  is  an  acute,  in  II,  an  obtuse,  in  III, 

72 


LAW  OF   SINES  — OF   TANGENTS. 


73 


a  right  angle.  The  demonstration  applies  to  each  triangle, 
but  in  II,  sin  A  CB=  sin  D  OB  (why?);  in  III,  sin  (7=1 
(why?). 


B 


b     D    C    A        b      C      D     A 
I.  II. 

FIG.  34. 


6 
III. 


Now 


p 

sin  A  —  — »  .*.  p  =  c  sin  A. 


sn  (= 

a 


Equating  values  of  p, 


or, 


/.  p  =  asm  0. 

c  sin  A  =  a  sin  (7, 
sin  A      sin  0 


o 


By  dropping  a  perpendicular  from  A,  or  (7,  on  a,  or  <?,  show 
that 


sin  B     sin  C 


-,  or 


sn 


whence 


sn 


sn 


a 
sin  (7 


60.  Law  of  Tangents.  The  tangent  of  half  the  difference 
of  two  angles  of  a  triangle  is  to  the  tangent  of  half  their  sum, 
as  the  difference  of  the  sides  opposite  is  to  their  sum. 

a  _  sin  A 
b      sin  B 


By  Art.  59, 


By  composition  and  division, 
a  —  b  _  sin  A  —  sin  B  _  2  cos  |  (A  + 


sn 


—  B} 


a  +  b      sin  A  +  sin  B  ~  2  sin 


cos 


-  B) 


or, 


tanj-04+.B)      a 


74 


PLANE   TRIGONOMETRY. 


61.  Law  of  Cosines.  The  cosine  of  any  angle  of  a  triangle 
is  equal  to  *the  quotient  of  the  sum  of  the  squares  of  the  adjacent 
sides  less  the  square  of  the  opposite  side,  divided  by  twice  the 
product  of  the  adjacent  sides. 


In  each  figure 


a=p 


(in  Fig.  34,  II,  DO  is  negative  ;  in  III,  zero) 


But 

AD  =  c  cos  A,     .'.  a2  = 


_  2  1  .  AD. 


Prove  that 
and 


cos^i  = 


cos  B  = 


cos     = 


2  be 


2ac 


2ab 


62.  Though  these  formulae  may  be  used  for  the  solution 
of  the  triangle,  they  are  not  adapted  to  the  use  of  loga- 
rithms (why?).  Hence  we  derive  the  following: 


Since     cos  A  =  2  cos2     -1  =  1- 

L 

we  have 


2cos24 


cos  A,  and  2  sin2  —  =  1  —  cos  A. 


< 


o 


LAW  OF   COSINES.  75 

From  the  latter 


9   i  2  A  __  -.  _  62  +  c2  -  a2      2  6g  -  52  -  c2  +  a2 


2  fo  2  be 

Let     a  +  5-f-0  =  2s,  then  a  +  5  —  c=a  +  b  +  c  —  2c=2s  —  2<?; 
t.«.  a  +  6  —  c=  2(s  —  c). 

In  like  manner,      «  —  b  +  <?  =  2  (s  —  5). 


Substituting, 


Show  that  sin  —  =  ? 

Zt 

O 
also  sin—  =  ? 


J- 

From  2  cos2  —  =  1  +  cos  A, 


£ 

show  that  cos  —  = 

7} 

also  cos  —  =  ? 

O 

and  cos  —  =  ? 

Also  derive  the  formulae 


tan-  =  ? 
tanf=? 


76  PLANE   TRIGONOMETRY. 

63.  Area  of  the  triangle.     In  the  figures  of  Art.  59  the 
area  of  the  triangle  ABC '=  A  =  %pb. 

But     p=csin.A.     .•.  A  =  ^bcsinA.  (i) 


Again,  by  law  of  sines,    b  =  — 

Substituting,  A  = 

z  sin  c/ 

-•—  J-~*  (why?)>          (ii) 


sin  O 

c2  sin  A  sin  B 
2  sin  0 

c^  sin  A  sin  B 


A       A 

Finally,  since  sin  A  =  2  sin  —  cos  —  ,  we  have  from  (i) 

" 


or  A  =  V*(*  —  a)(s  —  b)(s  —  c).        (iii) 

Find  A ;     (1)  Given  a  =  10,  b  =  12,  C  =  45°. 

(2)  Given  a  =  4,  6  =  5,  c  =  6. 

(3)  Given  a  =  2,  £  =  45°,  C  =  60°. 


SOLUTION  OF   TRIANGLES. 

64.   For  the  solution  of  triangles  we  have  the  following 
formulae,  which  should  be  carefully  memorized : 

y     sin  A  _  sin  B  _  sin  C 
a  b  c 

II.   tan*Ot-#)  =  ^^tanJ  (A  +  B). 
a  +  b 

III.    w»4  =  \/5z 


2  5c 

or  tan  ^  = 

IV.    A=J 


SOLUTION   OF  TRIANGLES.  77 

Which  of  the  above  forrnulse  shall  be  used  in  the  solution 
of  a  given  triangle  must  be  determined  by  examining  the 
parts  known,  as  will  appear  in  Art.  69.  It  is  always  pos- 
sible to  express  each  of  the  unknown  parts  in  terms  of  three 
known  parts. 

In  solving  triangles  such  as  Case  I,  Art.  58,  the  law  of 
sines  applies;  for,  if  the  given  side  is  not  opposite  either 
given  angle,  the  third  angle  of  the  triangle  is  found  from 
the  relation  A  +  B  +  C  =  180°,  and  then  three  of  the  four 

quantities  in  — —  =  — —  being  known,  the  solution  gives 

a  o 

the  fourth. 

In  Case  II  (5)  the  law  of  sines  applies,  but  in  II  (a)  two 

only  of   the   four  quantities   in  — —  =  — r —  are   known. 

a  b 

Therefore,  we  resort  to  the  formula 

tan  l  ( A  -  B)  =  ^~|  tan  %(A  +  £), 

in  which  all  the  factors  of  the  second  member  are  known. 
In  Case  III,   tan-  =  \K*~6)  ^s~~c^  is  clearly  applicable, 

O  II  o  /  o  n  \ 

£  c  ^o  —  Cl ) 

A  A 

and  is  preferred  to  the  formulae  for  sin—  and  cos  —  ;  for, 

—i  — 

first,  it  is  more  accurate  since  tangent  varies  in  magnitude 
from  0  to  oo,  while  sine  and  cosine  lie  between  0  and  1. 
(See  Art.  27,  5.) 

Let  the  student  satisfy  himself  on  this  point  by  finding,  correct  to 
seconds,  the  angle  whose  logarithmic  sine  is  9.99992,  and  whose  loga- 
rithmic tangent  is  1.71668.  Does  the  first  determine  the  angle  ?  Does 
the  second? 

And,  second,  it  is  more  convenient,  since  in  the  complete 

A 

solution  of  the  triangle  by  sin  —  six  logarithms  must  be  taken 

2 

A  A 

from  the  table,  by  cos  —  seven,  and  by  tan  —  but  four. 

2i  —. 

The  right  triangle  may  be  solved  as  a  special  case  by  the 
law  of  sines,  since  sin  (7=1. 


.0 


T8 


PLANE   TRIGONOMETRY. 


65.  Ambiguous  case.  In  geometry  it  was  proved  that  a 
triangle  having  two  sides  and  an  angle  opposite  one  of  them 
of  given  magnitude  is  not  always  determined.  The  marks 
of  the  undetermined  or  ambiguous  triangle  are : 

1.  The  parts  given  are  two  sides  and  an  angle  opposite  one. 

2.  The  given  angle  is  acute. 

3.  The  side  opposite  this  angle  is  less  than  the  other  given 
side. 

When  these  marks  are  all  present,  the  number  of  solutions 
must  be  tested  in  one  of  two  ways  : 

(#)  From  the  figure  it  is  apparent  that  there  will  be  no 
solution  when  the  side  opposite  is  less  than  the  perpendicular 
p  ;  one  solution  when  side  a  equals  p  ;  and  two  solutions  when 
a  is  greater  than  p. 


A  b  G    A  b  G  A  b      C 

No  Solution.  One  Solution.  Two  Solutions. 

FIG.  35. 

And  since  sin  A  —  —  ->  it  follows  that  there  will  be  no  solu- 

/> 

tion,  one  solution,  two  solutions,  according  as  sin  A  =  — 

<  c 

(b)  A  good  test  is  found  in  solving  by  means  of  loga- 
rithms ;  and  there  will  be  no  solutions,  one  solution,  two  solu- 
tions, according  as  log  sin  0  proves  to  be  impossible,  zero, 
possible,  i.e.  as  log  sin  G  is  positive,  zero,  or  negative.  This 
results  from  the  fact  that  sine  cannot  be  greater  than  unity, 
whence  log  sine  must  have  a  negative  characteristic,  or  be 
zero. 

66.  In  computations  time  and  accuracy  assume  more  than 
usual  importance.  Time,  will  be  saved  by  an  orderly  arrange- 
ment of  the  formulae  for  the  complete  solution,  before  open- 
ing the  book  of  logarithms,  thus  : 


SOLUTION  OF   TRIANGLES. 


79 


Given  A,  B,  a.     Solve  completely. 


(7=180°- 


6  = 


a  sin  j? 

sin^l  ' 


c=il»Lg   A=la6sin 

sin  .4. 


180° 
A+B  = 

log  a  =                                log  a  = 
log  sin  B  =                         log  sin  C  = 

loga  = 
log  6  = 
log  sin  C  = 
colog  2  = 

colog  sin  A  =                     colog  sin  A  = 

log  b  =                                 log  c  = 
.-.  6  =                                  .-.  c  = 
C^ecyt: 
log(s-6)  = 
log(s-c)  = 

COlog  8  = 

colog  (s  —  a)  = 

logA  = 

2) 

.•.A  = 


log  tan -  = 
.-.  ^  = 


67.  Accuracy  must  be  secured  by  checks  on  the  work  at 
every  step  ;   e.g.  in  adding  columns  of  logarithms,  first  add 
up,  and  then  check  by  adding  down.     Too  much  care  can- 
not  be   given   to  verification   in   the  simple  operations   of 
addition,  subtraction,  multiplication,  and  division.     A  final 
check  should  be  made  by  using  other  formulse  involving  the 
parts  in  a  different  way,  as  in  the  check  above.     As  far  as 
possible  the  parts  originally  given  should  be  used  through- 
out in  the  solution,  so  that  an  error  in  computing  one  part 
may  not  affect  later  computations. 

68.  The  formulsB  should  always  be  solved  for  the  unknown 
part  before  using,  and  it  should  be  noted  whether  the  solu- 
tion gives  one  value,  or  more  than  one,  for  each  part ;  e.g. 
the  same  value  of  sin  B  belongs  to  two  supplementary  angles, 
one  or  both  of  which  may  be  possible,  as  in  the  ambiguous 
case. 

69.  Write  formulae  for  the  complete  solution  of  the  fol- 
lowing triangles,  showing  whether  you  find  no  solution,  one 
solution,  two  or  more  solutions,  in  each  case,  with  reasons  for 
your  conclusion : 


80 


PLANE   TRIGONOMETRY. 


a            b 

c                       J. 

B                    C 

1. 

81°  26'  28" 

44°  11'  20"      54°  22'  12" 

2.                 78.54 

63°  18'  20" 

41°  30'  18" 

3.                135.82 

26.89             53°  28'  30" 

4.    0.75         0.85 

0.95 

5.     243 

562 

36°  15'  40" 

6.                 38.75 

25.92 

63°  50'  10" 

7.    0.058 

78°  15' 

33°  46' 

8.   2986 

1493 

30° 

9.                      48 

50 

26°  15' 

MODEL  SOLUTIONS. 

1.   Given  a  =  0.785,   b  =  0.85,   c  =  0.633.     Solve  completely. 

•f  r»  r-»                    *%  /  ^                '  ^ 

"*          -f  o  -r»            —  *\/  ^              • 

C     J(s-a)(s-b) 

tdll     _.      V                .                 ^ 

2              s(s  —  a) 
Check:  A  +  B 

IU2       *      s(s-b) 

'U  2       *      s(s-c) 

+  C  =  180°.     A  =  Vs(s  - 

aXs-b)(s-c). 

a  =  0.735 
b  =  0.85 
c  =  0.633  , 

log  (*-&)  =    9.45332 
log(t-c)  =    9.69984 
colog  s  =    9.94539 
colog  (s  -  a)  =  .  0.45717 

log(s-a)=    9.54283 
log  (s-  c)=    9.69984 
colog  s=    9.94539 
colog  (s-  ft)  =    0.54668 

2)2.268 
s  =  1.134 

2)19.55572 

2)19.73474 

s  -  a  =  0.349 

log  tan  £4=    9.77786 

log  tan  £  B  =    9.86737 

s  -  b  =  0.284 
«  -  c  =  0.501 

\A  =30°  56'  49" 
A  =  61°  53'  38" 

$  5  =  36°  23'  2" 
£  =  72°  46  '4" 

CAecfc: 
.4  =    61°  53'  38" 
B=    72°  46'    4" 
C=:    45°  20'  20" 

log(*-a)=    9.54283 
log  («-&)  =    9.45332   ' 
colog  s=    9.94539 

r»r»lno'  fe          s<\            0  ^001  fi 

log  5  =    0.05461 
log  (<?-«)=    9.54283 
log  («-&)=    9.45332 
log  (s  -  c)  =    9.69984 

COlOg  ^o   —   Cj  —      U.OUU1D 

2)18.75060 
log  A  =    9.37530 
A=    0.2373 

180°   0'   2" 

2)19.24170 
log  tan  \  C  =    9.62085 

\  C  =  22°  40'  10" 
C  =  45°  20'  20" 

Given  a  =  30,  b^  40,  c  =  50. 
\(2)  Given  a  =  2159,  &  =  1431.6,  c  =  914.8. 
(3)  Given  a  =  78.54,  b  =  32.56,  c  =  48.9. 


SOLUTION  OF  TRIANGLES. 


81 


2.   Given  A=o7°2&  12",  C  =  68°  15'  30",  c  =  832.56.    Solve  completely. 

_  c  sin  A 

sin  C  sin  C 

£  =  180° -(4  +  C) 


=  54°  21'  18". 


Check:  tan  %  A  = 


log  c  =  2.92042  log  c  =  2.92042 

log  sin  A  =  9.92548          log  sin  B  =  9.90990 

colog  sin  C  =  0.03204      colog  sin  C  =  0.03204 


s  (s  —  a) 

log  b  =    2.86236 

logc=    2.92042 

log  sin  A  =    9.92548 


log  a  =  2.87794 
a=    754.98 

Check :  a  =  754.98 
6  =  728.38 
c  =  832.56 


log  b  =  2.86236  log  2  A  =    5.70826 

b  =    728.38     A  =  510811  =  255405.5 


2)2315.92 
s  =  1157.96 


s-a=    402.98        log  (*  -  fe)  =  2.63304 

s  -  b  =    429.58        log  (J  -  c)  =  2.51242 

s-c=    325.40               colog  *=  6.93634 

colog  (s-  a)  =  7.39471 

2)19.47651 

log  tan  1 .4  =    9.73826 
\A  =28°  41' 38" 
A  =  57°  23'  16" 
Solve : 

(1)  Given  a  =  215.73,  B  =  92°  15',  C  =  28°  14'. 
•2)  Given  b  =  0.827,  A  =  78°  14'  20",  B  =  63°  42'  30". 
Q  (3)  Given  6  =  7.54,  c  =  6.93,  £  =  54°  28'  40". 

3.    Given  a  =  25.384,  c  =  52.925,  B  -  28°  32'  20".     Solve  completely. 
(Why  not  use  the  same  formulae  as  in  Example  1,  or  2?) 
C  - 


tan 


c  —  a 


tan 


2          c  +  a  "'         2  sin  C 

180°  -  B  =  C  +  A  =  151°  27'  40". 
.-.  l(C  +  4)=    75°  43' 50". 

c=  52.925  log  (c-a).=  1.43998 

a=  25.384         colog  (c  +  a)  =8.10619 


,    A  =  lac  sin 5. 


6  = 


a  sin  B 
sin  -^4 


i(c-^)=  54°  7'38" 
l(C+4)=  75°43'50" 


c-a=  27.541  log  tan  1(C  --4)  =  0.14077  subtracting,  .4=  21°36'12" 

log  a  =  1.40456 


log  c  =  1.72366 

log  sin  5  =  9.67921 

colog  sin  C  =  0.1 1484 

log  b  =  1.51771 
b=  32.939 


:  log  a  =  1.40456 
log  sin  £  =  9.67921 
colog  sin  A  =0.43395 

log  b  =  1.51772 


log  c  =  1.72366 
log  sin  5  =  9.67921 

log  2  A  =  2.70743 


£2  PLANE   TRIGONOMETRY. 

Solve  :\1)    Given  a  =  0.325,    c  =  0.426,    B  =  48°  50'  10". 
*(2)    Given   b  =  4291,     c  =  3194,     A  =  73°  24'  50".    - 
(3)    Given  b  =  5.38,     c  =  12.45,    A  =  62°  14'  40". 

4.  Ambiguous  cases.  Since  the  required  angle  is  found 
in  terms  of  its  sine,  and  since  sin  a  =  sin  (180°  —  a),  it  fol- 
lows that  there  may  be  two  values  of  a,  one  in  the  first,  and 
the  other  in  the  second  quadrant,  their  sum  being  180°.  In 
the  following  examples  the  student  should  note  that  all  the 
marks  of  the  ambiguous  case  are  present.  The  solutions  will 
show  the  treatment  of  the  ambiguous  triangle  having  no 
solution,  one  solution,  two  solutions. 

O)  Given  b  =  70,  c  =  40,  0=  47°  32'  10".  Solve.  Why 
ambiguous  ? 

.     p      b  sin  C  log  b  =1.84510 

c  log  sin  0  =  9. 86788 

cologc  =  8.39794 

log  sin  .£  =  0.11092 

.•.  B  is  impossible,  and  there  is  no  solution.  Why? 
Show  the  same  by  sin  0>j" 

(5)  Given  a  =  1.5,  c  =  1.7,  A  =  61°  55'  38".     Solve. 

.    „     csiuA  log  c=  0.23045 

a  log  sin  A=  9.94564 

cologa=  9.82391 

log  sin  C  =0.00000 
(7=90° 

and    there   is   one   solution.     Why?     Show   the   same    by 

sin  A  =  -.      Solve  for  the  remaining  parts  and  check   the 

o 
work. 


SOLUTION  OF   TRIANGLES.  83 

. 
O)  Given  a  =  0.23$,  b  =  0.18J),  B  =  36°  28'  20".     Solve. 

a  sin  B  b  sin  (7 

sinJ.  =  — — — ,  c  = --. 

o  sin  .D 

log  a  =  9.3710T  log  b  =  9.27646.        9.27646 

log  sin  5=9. 77411  log  sin  (7=9. 99772  or  9. 28774 

colog  6  =  0.72354  colog  sin  B  =  0.22589        0.22589 

log  sin  A  =  9. 86872  log  <?  =  9.50007  or  8.79009 

A  =  47°  39'  25"  c  =  0.31628  or  0.06167 

or  132°  20'  35". 

...  (7=95°  52'  15"  or  11°  11'  5". 

Solve  for  A,  and  check.     Show  the  same  by  sin  B  <  — 

a 

Solve  : 

(1)  Given  b  =  216.4,  c  =  593.2,  B=  98°  15'. 

(2)  Given  a  =  22,  5  =  75,  5  =  32°  20'. 

(3)  Given  a  =  0.353,  c  =  0.295,  A  =  46°  15'  20". 
^(4)  Given  a  =  293.445,  b  =  450,  ^  =  40°  42'. 

Q  (5)  Given  b  =  531.03,  c  =  629.20,  J5  =  34°  28'  16". 

Solve  completely,  given  : 

a  b  c  A  B  C 

1.  50  60  78°  27'  47" 

2.  10  11  93°  35' 

3.  4  5  6 

10  109°  28' 16"    38°  56' 54" 

5.           40           51  49°  28' 32" 

352.25    513.27    482.68 

7.       0.573      0.394  112°   4' 

1*.   107.087  56°  15'           48°  35' 

V2  117°                45° 

t  10.     197.63    246.35  34°  27' 
-11.        4090       3850       3811 

-fj.2.       3795  73°  15' 15"    42°  18' 30" 

• — -  13.                     234.7      185.4  84°  36' 

014.                   26.234    22.6925  49°   8' 24" 

k   15.         273         136  72°  25' 13" 


84  PLANE  TRIGONOMETRY. 

APPLICATIONS. 

70.  Measurements  of  heights  and  distances  often  lead  to 
the  solution  of  oblique  triangles.  With  this  exception,  the 
methods  of  Chapter  V  apply,  as  will  be  illustrated  in  the 
following  problems. 

The  bearing  of  a  line  is  the  angle  it  makes  with  a  north 
and  south  line,  as  determined  by  the  magnetic  needle  of  the 
mariner's  compass.  If  the  bearing  does  not  correspond  to 
any  of  the  points  of  the  compass,  it  is  usual  to  express  it 
thus:  N.  40°  W.,  meaning  that  the  line  bears  from  N.  40° 
toward  W. 

EXAMPLES. 

1.  When  the  altitude  of  the  sun  is  48°,  a  pole  standing  on  a  slope 
inclined  to  the  horizon  at  an  angle  of  15°  casts  a  shadow  directly  down 
the  slope  44.3  ft.  How  high  is  the  pole? 

N  2.  A  tree  standing  on  a  mountain  side  rising  at  an  angle  of  18°  30' 
breaks  32  ft.  from  the  foot.  The  top  strikes  down  the  slope  of  the  moun- 
tain 28  ft.  from  the  foot  of  the  tree.  Find  the  height  of  the  tree. 

3.  From  one  corner  of  a  triangular  lot  the  other  corners  are  found  to 
be  120  ft.  E.  by  N.,  and  150  ft.  S.  by  W.     Find  the  area  of  the  lot,  and 
the  length  of  the  fence  required  to  enclose  it.  jp  t  £ 'Is ,  A  ~?  t>'  4-  *£" 

4.  A  surveyor  observed  two  inaccessible  headlands,  A  and  B.     A  was 
W.  by  N.  and  B,  N.E.     He  went  20^miles  N.,  when  they  were  S.W.  and 
S.  by  E.     How  far  was  A  from  B  ?  ?*t/ou>* 

5.  The  bearings  of  two  objects  from  a  ship  were  N.  by  W.  and  N.E. 
by  N.     After  sailing  E.  11  miles,  they  were  in  the  same  line  W.N.W. 
Find  the  distance  between  them. 

6.  From  the  top  and  bottom  of  a  vertical  column  the  elevation  angles 
of  the  summit  of  a  tower  225  ft.  high  and  standing  on  the  same  hori- 
zontal plane  are  45°  and  55°.     Find  the  height  of  the  column. 

""  7.  An  observer  in  a  balloon  1  mile  high  observes  the  depression  angle 
of  an  object  on  the  ground  to  be  35°  20'.  After  ascending  vertically  and 
uniformly  for  10  mins.,  he  observes  the  depression  angle  of  the  same  object 
to  be  55°  40'.  Find  the  rate  of  ascent  of  the  balloon  in  miles  per  hour. 
—  8.  A  statue  10  ft.  high  standing  on  a  column  subtends,  at  a  point 
100  ft.  from  the  base  of  the  column  and  in  the  same  horizontal  plane,  the 
same  angle  as  that  subtended  by  a  man  6  ft.  high,  standing  at  the  foot 
of  the  column.  Find  the  height  of  the  column.  *"[(•>  »  Q~ O  2> 

9.   From  a  balloon  at  an  elevation  of  4  miles  the  dip  of  the  horizon  / 
is  2°  33'  40".     Required  the  earth's  radius. 

fev 


TRIANGLES  —  APPLICATIONS.  85 

#*—  7  H  *  r6   T£  *  t~jt 

10.   Two  ships  sail  from  Boston,  one  S.E.  50  miles,  the  other  N.E.  by 
E.  60  miles.     Find 

first. 


,  ..  ,  .. 

ind  the,  bearing  and  distance  of  the  second  ship  from  the 

O,  *f  6,  A  =    S  6T6  *  6;<?  -  »„  "/&'/*" 


11.  The  sides  of  a  valley  are  two  parallel  ridges  sloping  at  an  angle  of 
30°.  A  man  walks  200  yds.  up  one  slope  and  observes  the  angle  of  eleva- 
tion of  the  other  ridge  to  be  15°.  Show  that  the  height  of  the  observed 
ridge  is  273.2  yds. 

O  12.  To  determine  the  height  of  a  mountain,  a  north  and  south  base 
line  1000  yds.  long  is  measured  ;  from  one  end  of  the  base  line  the  sum- 
mit  bears  E.  10°  N.,  and  is  at  an  altitude  of  13°  14'.  From  the  other  end 
it  bears  E.  46°  30'  N.  Find  the  height  of  the  mountain. 

13.  The  shadow  of  a  cloud  at  noon  is  cast  on  a  spot  1600  ft.  due  west 
of  an  observer.     At  the  same  instant  he  finds  that  the  cloud  is  at  an  ele- 
vation of  23°  in  a  direction  W.  14°  S.     Find  the  height  of  the  cloud  and 
the  altitude  of  the  sun. 

14.  From  the  base  of  a  mountain  the  elevation  of  its  summit  is  54°  20'. 
From  a  point  3000  ft.  toward  the  summit  up  a  plane  rising  at  an  angle 

of  25°  30'  the  elevation  angle  is  68°  42'.    Find  the  height  of  the  mountain.        ^ 
•~  15.    From  two  observations  on  the  same  /xV 

meridian,   and  92°  14'   apart,   the   zenith  ° 

.angles  of  the  moon   are   observed   to   be 
44°  54'  21"    and   48°  42'  57".      Calling   the 

earth's  radius  3956.2  miles,  find  the   dis- 

k~.  __  y\Z-  Zenith  angle 
tance  to  the  moon.  I  —U. 

16.  The  distances  from  a  point  to  three 
•objects  are  1130,  1850,  1456,  and  the  angles 

subtended  by  the  distances  between  the  three  objects  are  respectively 
102°  10',  142°,  and  115°  50'.    Find  the  distances  between  the  three  objects. 

17.  From  a  ship  A  running  N.E.  6  mi.  an  hour  direct  to  a  port  dis- 
tant 35  miles,  another  ship  B  is  seen  steering  toward  the  same  port,  its 
bearing  from  A  being  E.S.E.,  and  distance  12  miles.     After  keeping  on 
their  courses  1$  hrs.,  B  is  seen  to  bear  from  A  due  E.     Find  B's  course 
and  rate  of  sailing. 

18.  From  the  mast  of  a  ship  64  ft.  high  the  light  of  a  lighthouse  is 
just  visible  when  30  miles  distant.     Find  the  height  of  the  lighthouse, 
the  earth's  radius  being  3956.2  miles. 

19.  From  a  ship  two  lighthouses  are  observed  due  N.E.     After  sailing 
20  miles  E.  by  S.,  the  lighthouses  bear  N.N.W.  and  N.  by  E.     Find  the 
distance  between  the  lighthouses. 

20.  A  lighthouse  is  seen  N.  20°  E.  from  a  vessel  sailing  S.  25°  E.     A 
mile  further  on  it  appears  due  N.     Determine  its  distance  at  the  last 
observation. 


EXAMPLES   FOR   REVIEW. 

IN  connection  with,  each  problem  the  student  should  review 
all  principles  involved.  The  following  list  of  problems  will  then 
furnish  a  thorough  review  of  the  book.  In  solving  equations, 
find  all  values  of  the  unknown  angle  less  than  360°  that  satisfy 
the  equation. 

1.  If  tan  «  =  f,  tan  ft  =  |,  show  that  tan  (ft  -  2  a)  =  T2T. 

2.  Prove  tan  a  +  cot  a  =  2  esc  2  a. 

3.  From  the  identities  sin2^  +  cos2^-  =  1,  and  2  sin ^  cos—  =  sin  A, 


.    A 


prove  2  sin  —  =  ±  Vl  +  siii  A  ±  Vl  -  sin  A, 


and  2  cos  —  =  ±  Vl  +  sin  A  T  Vl  -  sin  4. 

4.  Remove  the  ambiguous  signs  in  Ex.  3  when  A  is  in  turn  an  angle 
of  each  quadrant. 

5.  A  wall  20  feet  high  bears  S.  59°  5'  E. ;  find  the  width  of  its  shadow 
on  a  horizontal  plane  when  the  sun  is  due  S.  and  at  an  altitude  of  60°. 

6.  Solve  sin  x  +  sin  2  x  +  sin  3  x  =  1  +  cos  x  +  cos  2  x. 

7.  Prove  tan-1  -  +  tan-1  -  =  -• 

8.  If  A  =  60°,  B  =  45°,  C  =  30°,  evaluate 

tan  A  +  tan  B  +  tan  C 


tan  A  tan  B  +  tan  B  tan  C  +  tan  C  tan  A 

9     Prove  cos  Q4  +  B)  cos  C  _  1  -  tan  A  tan  £ 
cos  (A  +  C)  cos  fi  ~  1  -  tan  ^4  tan  C* 

10.    Solve  completely  the  triangle  whose  known  parts  are  b  =  2.35, 
=  1.96,  C  = 


11.  Find  the  functions  of  18°,  36°,  54°,  72°. 

Let  x  =  18°.    Theii  2  x  =  36°,  3x  =  54°,  and  2  x  +  3  x  =  90°. 

12.  If  cot  a  =  -,  find  the  value  of 

sin  «  +  cos  a  +  tan  a  +  cot  a  +  sec  a  +  esc  a. 


EXAMPLES   FOR  REVIEW.  8T 

13.  Prove  sin  ««  sin  2  g  -  sin  3  ,8  sin  2  «  =  j  +  4  cos  «  cos  ^ 

sm  2  a  sin  /?  —  sin  2  /?  sin  a 

14.  From   a  ship  sailing   due  N.,  two  lighthouses   bear  N.E.  and 
N.N.E.,  respectively;   after  sailing  20  miles  they  are  observed  to  bear 
due  E.     Find  the  distance  between  the  lighthouses. 

15.  Solve  1-2  sin  x  =  sin  3  x. 

16.  Prove  sin-K/—  ^-=  tan 

*a  +  b 

17.  If  cos  0  -  sin  0  =  V2  sin  0,  then  cos  0  +  sin  0  =  V2  cos  0. 

18.  Solve  completely  the  triangle  ABC,  given  a  =  0.256,  b  =  0.387r 
C  =  102°  20'.5. 

2  cos  2  <*  ~  * 


19.  Prove  tan  (30°  +  a)  tan  (30°-  a)  = 

2  cos  2  a  +  1 

20.  Solve  tan  (45°  -  0)  +  tan  (45°  +  0)  =  4. 

21.  Prove  sin2  «  cos2  ft  -  cos2  a  sin2  (3  =  sin2  a  -  sin2  ft. 

22.  Prove  cos2  a  cos2  /?  -  sin2  a  sin2  /?  =  cos2  a  -  sin2  /?. 

23.  A  man  standing  due  S.  of  a  water  tower  150  feet  high  finds  its- 
elevation  to  be  72°  30'  ;  he  walks  due  W.  to  A  street,  where  the  elevation 
is  44°  50'  ;  proceeding  in  the  same  direction  one  block  to  B  street,  he  finds 
the  elevation  to  be  22°  30'.     What  is  the  length  of  the  block  between  A 
and  B  streets  ? 

24.  Prove  tan-1  -  +  tan-1  -  +  tan-1  -  +  tan-1  -  =  -• 

3  5  7  84 

25.  If  P  =  60°,  Q  =  45°,  R  =  30°,  evaluate 

sin  P  cos  Q  +  tan  P  cos  Q 
sin  P  cos  P  +  cot  P  cot  R 

26.  If  cos  (90°  +  «)  =  -£,  evaluate  3  cos  2  a  +  4  sin  2  a. 

27.  If  sin  B  +  sin  C  =  m,  cos  5  +  cos  C  =  n,  show  that  tan^jt-S-  -. 

2  n 

28.  Show  that  sin  2  /?  can  never  be  greater  than  2  sin  /?. 

29.  Prove  sin-1  f  +  sin-1  T57  =  tan-1  f  f  . 

30.  Solve  cot-1*  +  sin-i-VS  =  £• 

5  4 

31.  Solve  sin-1  a;  +  sin-1(l  —  x)=  cos-1*. 

32.  A  man  standing  between  two  towers,  200  feet  from  the  base  of 
the  higher,  which  is  90  feet  high,  observes  their  altitudes  to  be  the  same  ; 
70  feet  nearer  the  shorter  tower  he  finds  the  altitude  of  one  is  twice  that 
of  the  other.     Find  the  height  of  the  shorter  tower,  and  his  original 
distance  from  it. 


88  PLANE   TRIGONOMETRY. 

33.  Solve  cos  3  /?  +  8  cos8  ft  =  0. 

34.  Solve  cot  m  -  tan  (180°  +  m)  =  sec  m  +  sec  (90°  -  m). 


35.  Solve      ~  =  2cos2*. 

1  +  tan  t 

36.  Prove  cot  4  +  cotjB  =  sin 


sin  A  sin  .B 

37.  Prove  cot  P  -  cot  Q  =  -  sm(p~  Q). 

sm  P  sin  Q 

38.  In  the  triangle  ABC  prove 

a  =  b  sin  C  +  c  sin  5, 
6  =  c  sin  vl  -f  a  sin  C, 
c  =  a  sin  B  +  &  sin  ^4. 

39.  Solve  completely  the  triangle,  given 

a  =  927.56,  b  =  648.25,  c  =  738.42. 

40.  Prove  cos2  a  -  sin  (30°  +  «)  sin  (30°  -  a)  =  f  . 

41.  Prove  tan  3  x  tan  x  =  COS  2  *  ~  COS  4  *• 

cos  2  a;  -f  cos  4  # 

42.  Simplify  cos  (270°  +  «)  +  sin  (180°  +  «)  +  cos  (90°  +  a). 

43.  Simplify  tan  (270°  -0}-  tan  (90°  +  6}  +  tan  (270°  +  0). 

44.  Solve  cos  3  <£  —  cos  2  <£  +  cos  <£  =  0. 

45.  Solve  cos  A  -f  cos  3  A  +  cos  5  A  +  cos  7  A  =0. 

46.  The  topmast  of  a  yacht  from  a  point  on  the  deck  subtends  the 
same  angle  «,  that  the  part  below  it  does.     Show  that  if  the  topmast  be 
a  feet  high,  the  length  of  the  part  below  it  is  a  cos  2  a. 

47.  A  horizontal  line  AB  is  measured  400  yards  long.     From  a  point 
in  AB  a  balloon  ascends  vertically  till  its  elevation  angles  at  A  and  B 
are  64°  15'  and  48°  20;,  respectively.     Find  the  height  of  the  balloon. 

48.  If  cos  <j>  =  n  sin  cc,  and  cot  <£  =  —  —  ,  prove  cos  y8  =  -  —  . 

tan/3  VI  +  n2  cos2  a 

49.  Find  cos  3  cc,  when  tan  2  a  =  —  f  . 

50.  Solve    completely  the  triangle,  given   a  =  0.296,   5  =  28°  47  '.3, 
C  =  84°  25'. 

51.  Evaluate  sin  300°  +  cos  240°  +  tan  225°. 

52.  Evaluate  sec  ^  -  esc  ^  +  tan  ^ 

o  o  o 


EXAMPLES   FOR   REVIEW.  89 

53.  If  fon  0  =  sin  «  cosy- sin  ff  sin  y 

cos  a  cos  y  —  cos  ft  sin  y 

,       sin  a  sin  y  —  sin  B  cos  y 

and  tan  <£  =  —          — «-—  — *> 

cos  «  sin  y  —  cos  p  cos  y 

show  that  tan(0  +  <£)  =  tan(a  +  (3). 

54.  If  tan  466°  15'  38"  =  -  ty,  find  the  sine  and  cosine  of  233°  7'  49". 

esc  a  —  cot  a     sec  a  —  tan  a 

55.  Prove  = 

sec  a  +  tan  a     esc  a  +  cot  a 

56.  Prove  cos(«  -  3fl)- cos(3  «  -  ft)  =  2  sin(a  _  ^ 

sin  2  a  +  sin  2  /2 

57.  Prove  sin  80°  =  sin  40°  +  sin  20°. 

58.  Prove  cos  20°  =  cos  40°  +  cos  80°. 

59.  Prove  4  tan-1  -  -  tan"1  —  =  -• 

5  239      4 

60.  From  the  deck  of  a  ship  a  rock  bears  N.N.W.     After  the  ship 
has  sailed  10  miles  E.N.E.,  the  rock  bears  due  W.     Find  its  distance 
from  the  ship  at  each  observation. 

61.  Find  the  length  of  an  arc  of  80°  in  a  circle  of  4  feet  radius. 

62.  Given  tan  6  =  f ,  tan  <£  =  T\,  evaluate  sin(0  +  <£)  +  cos(0  —  </>). 

63.  If  tan  0  =  2  tan  <£,  show  that  sin (0  +  <£)  =  3  sin(0  -  <£). 

64.  Prove  cos(a  +  j8)cos(«-)8)+sin(a  +  ^)sin(a-/8)=l: 

65.  Solve  4  cos  2  0  +  3  cos  0  =  1. 

66.  Solve  3  sin  a  =  2  sin  (60°  -  «). 

67.  Prove  (sin  a  -  esc  «)2  -  (tan  a  -  cot  a)2  +  (cos  a  -  sec  a)2  =  1. 

68.  Prove  2 (sin6  a  +  cos6  a)  +  1  =  3 (sin4  a  +  cos4  a). 

69.  Prove  esc  2  /?  +  cot  4  /?  -  cot  /3  -  esc  4  /3. 

70.  If  tan  jt?  =  A,  cos  2  9  =  5?Z,  then  esc  £-=^2  =  5  Vl3. 

12  6.2o  2 

71.  Solve  completely  the  triangle,  given 

a  =  0.0654,   6  =  0.092,   £  =  38°40'.4. 

72.  Solve  completely  the  triangle,  given 

b  =  10,   c  =  26,   B  =  22°  37'. 

73.  A  railway  train  is  travelling  along  a  curve  of  £  mile  radius  at  the 
rate  of  25  miles  per  hour.     Through  what  angle  (in  circular  measure) 
will  it  turn  in  half  a  minute  ? 


90  PLANE  TRIGONOMETRY. 

74.  Express  the  following  angles  in  circular  measure  : 

63°,     4°  30',     6°  12'  36". 

75.  Express  the  following  angles  in  sexagesimal  measure  : 

7T        3_7T        17  7T 

6'       8  '       64  ' 

76.  If  A,  B,  C  are  angles  of  a  triangle,  prove 

A        PI       r 

cos  A  +  cos  B  +  cos  C  =  I  +  4  sin  —  sin  -  sin  -• 

77.  Prove  sin  2  x  +  sin  2  y  +  sin  2  2  =  4  sin  x  sin  y  sin  2,  when  or,  y,  z 
are  the  angles  of  a  triangle. 

78.  Prove  sec  a  =  1  +  tan  a  tan  -• 

79.  Prove  sin2  (a  +  /?)  -  sin2  (a  -  /?)  =  sin  2  a  sin  2  £. 

80.  Prove  cos2  (a  +  /?)  -  sin2  (a  -  /?)  =  cos  2  «  cos  2  /?. 

81.  Prove  sin  19  p  +  sin  17  p  =  2  cog  g 

sin  10  p  +  sm  Sp 

82.  Consider  with  reference  to  their  ambiguity  the  triangles  whose 
known  parts  are  : 

(a)  a  =  2743,  b  =  6452,  B  =  43°  15'  ; 

(6)  a  =  0.3854,  c  =  0.2942,  C  =  38°20'; 

(c)  b=    5,  c  =  53,  £  =  15°22'; 

(d)  a  =  20,  ft  =  90,  ^  =  63°  28'.5. 

83.  From  a  ship  at  sea  a  lighthouse  is  observed  to  bear  S.E.     After 
the  ship  sailed  N.E.  6  miles  the  bearing  of  the  lighthouse  is  S.  27°  30'  E. 
Find  the  distance  of  the  lighthouse  at  each  time  of  observation. 


84.  Prove  -KJ  +  8*)+rin(8tf  +  fl  =  2  cos  (0  +  #. 

sin  2  6  +  sin  2  <f> 

85.  Prove  cos  15°  -  sin  15°  =  -L- 

V2 

86.  Show  that  cos  («  +  /?)  cos  (a  -  /?)  =  cos2  a  -  sin2  /J 

=  cos2  ft  —  sin2  a. 

87.  Show  that  tan  (a  +  45°)  tan  («  -  45°)  =  ^  sin^  ~  *• 

88.  Solve  sin  (x  -f  y)  sin  (x  —  y}=  ^,    cos  (#  -f  y)  cos  (x  —  y)  =  0. 

89.  Prove  X  +  sin  *  ~  COS  a  =  tan  «. 

1  +  sin  a.  +  cos  «  2 


EXAMPLES  FOR  REVIEW.  91 

90.  Prove  tan  2  $  +  sec  2  6  =  COS  *  +  shl  £ 

cos  0  —  sm  0 

91.  If  tan  <#>=-,  then  a  cos  2  </>  +  &  sin  2  <£  =  a. 

92.  Prove  sin-1—  +  cot-1 3  =  £• 

V5  ^ 

93.  Solve  cos  A  +  cos  7  A  =  cos  4:  A. 

94.  Two  sides  of  a  triangle,  including  an  acute  angle,  are  5  and  7, 
the  area  is  14 ;  find  the  other  side. 

95.  Show  that  3co33fl-2cose-cos50  =  ^ 2  ft 

sin  5  6  —  3  sin  3  0  +  4  sm  0 

96.  A  regular  pyramid  stands  on  a  square  base  one  side  of  which  is 
173.6  feet.     This  side  makes  an  angle  of  67°  with  one  edge.     What  is 
the  height  of  the  pyramid  ? 

97.  From  points  directly  opposite  on  the  banks  of  a  river  500  yards 
wide  the  mast  of  a  ship  lying  between  them  is  observed  to  be  at  an  eleva- 
tion of  10°  28'.4  and  12°  14'.5,  respectively.     Find  the  height  of  the  mast. 

98.  Show  that  (sin  60°  -  sin  45°)  (cos  30°  +  cos  45°)  =  sin2  30°. 

99.  Find  x  if  sin-1  x  +  sin-1  x  =  |- 

100.   Trace  the  changes  in  sign  and  value  of  sin  a  -f  cos  a  as   a 
changes  from  0°  to  360°. 


CHAPTER    VIII. 


MISCELLANEOUS    PROPOSITIONS. 

71.  The  circle  inscribed  in  a  given  triangle  is  often  called 
the  incircle  of  the  triangle,  its  centre  the  incentre,  and  its 
radius  is  denoted  by  r.  The  incentre  is  the  point  of  inter- 
section of  the  three  bisectors  of  the  angles  of  the  triangle 
(geometry). 

The  circle  circumscribed  about  a  triangle  is  called  the 
circumcircle,  its  centre  the  circumcentre,  and  its  radius  R. 
The  circumcentre  is  the  point  of  intersection  of  perpendicu- 
lars erected  at  the  middle  points  of  the  three  sides  of  the 
triangle  (geometry). 


Incircle. 


The  circle  which  touches  any  side  of  a  triangle  and  the 
other  two  sides  produced  is  called  the  escribed  circle;  its 
radius  is  denoted  by  ra,  rb,  or  rc,  according  as  the  escribed 
circle  is  opposite  angle  J.,  J5,  or  C. 

Again,  the  altitudes  from  the  vertices  of  a  triangle  meet 
in  a  point  called  the  orthooentre  of  the  triangle. 

Finally,  the  medians  of  a  triangle  meet  in  a  point  called 
the  centroid,  which  is  two-thirds  of  the  length  of  the  median 
from  the  vertex  of  the  angle  from  which  that  median  is 
drawn  (geometry). 

Certain  properties  of  the  above  will  now  be  considered. 

92 


MISCELLANEOUS  PROPOSITIONS. 


93 


72.   To  find  the  radius  of  the  incircle. 

Let  A,  A',  A,"  A'"  represent 
the  areas  of  triangles  ABC, 
COB,  AOC,  BOA,  respectively. 

Then 

A  =  A'  +  A"  +  A'" 
=  \ |  (a  +  b  +  c)  r  =  sr. 


c 
FIG.  38. 


And  since      A  =  Vs(s  -  a)(*  — 


(Art.  63) 


COR.     To  express  the  angles  in  terms  of  r  and  the  sides, 
divide  each  member  of  the  above  equation  by  s  —  a. 


A      VJO-  a)(s-  ft)(t-0) 


Then 


—  a 


«(«  —  a 


tan  |-  A.        (Art.  62) 


In  like  manner   tan  £  B  =  —  -  —  ;    tan  4  0  —  —  - 


s  —  b 
To  find  the  radius  of  the  circumcircle. 


c 


s  —  c 


FIG.  39. 

In  the  figure  AB  C  is  the  given  triangle,  and  A'  C  a  diam 
eter  of  the  circumcircle.     Then,  angle  A  =  A',  or  180°  —  Ar 

.*.  sin.  A  =  sin.4/. 
Since  ABO  is  a  right  angle, 


_£a_ 
sin  A 


94 


PLANE   TRIGONOMETRY. 


COR.  1.     Asabove, 


sin  A     sin  B     sin  C 
another  proof  of  the  "law  of  sines." 

a 


,,  which  is 


COB.  2.     From  R  = 


•,  we  have 

2  sin  A 

cibc  cibc      -i  .  TWV 

— -  =  _ — ,  where  A=  area  ABO. 
ZbcswA     4  A 


74.   To  find  the  radii  of  the  escribed  circles. 


Represent  areas  ABC,  BOA, 
AGO,  BOG,  by  A,  A',  A",  A'", 
respectively.  Then  ra  is  the 
altitude  of  each  of  the  triangles 
BOA,AOC,BOC. 


—  a 
A 


In  like  manner,    rb  =  -    —  , 
s  —  o 


s  —  c 


75.   The  orthocentre. 

Denote  the  perpendiculars  on  the  sides 
a,  b,  c,  by  APa,  BPb,  OPC,  and  let  it  be 
required  to  find  the  distances  from  their 
intersection  0  to  the  sides  of  the  tri- 
angle, and  also  to  the  vertices. 

=  APht&nOAO. 


But 


cosA,  and  CA  0  =  90°  -  C. 


.-.   OPb  =  ccosAcotC  = 
=  2  R  cos  A  cos  O. 


sin  (7 


cos  A  cos  O. 

(Art.  73,  Cor.  1) 


MISCELLANEOUS   PROPOSITIONS.  95 

In  like  manner,  OPC  =  2  R  cos  B  cos  A, 
2RcosC  cos  B. 


APb         c  cos  A 


Again,  the  distances  from  the  orthocentre  to  the  vertices 
are, 

OA  = 


cos  OA  0       sin  0 
—  2  R  cos  A. 

Also,  OB  =  2  R  cos  B, 

and  OO=  2  R  cos  C. 

76.  Centroid  and  medians. 

The  lengths  of  the  medians  may  be  computed  as  follows : 

In  the  figure  the  medians  to  the 
sides  a,  b,  c,  are  AMa,  BMb,  CMC, 
meeting  in  the  centroid  0. 

Now,  by  the  law  of  cosines,  from  ^^^   ^\    ^^~^g 

the  triangle  BMbC, 

FIG.  42. 

BW  =  *+     _ 

=  a2  4-  —  —  ab  cos  (7. 

O        .        TO  O 

But,  cos  (7  = 


''lab 


whence,  BMb  =  i  V2  «2+  2  c2  -  b*  =  £  V«2  +  c2  +  2 

«2  +  6<2_52 

since  —  =  cos  B. 

2ac 

In  like  manner, 


OMC  =  1 V2  J2  +  2  a2  -  c8  =  J  V&2  +  a2  +  2  fta 


and          ^4Jf  =     V2  &  +  2  52  -  a2  =     Vc2  +  b2  +  2  c?6 


96  PLANE   TRIGONOMETRY. 

EXAMPLES. 

1.  In  the  triangle,  a  =  25 ,  6  =  35,  c  =  45,  find  R,  r,  ra. 

2.  Given   a  —  0.354,  b  =  0.548,  C  =  28°  34'  20",  find  the  distances  to 
C  and  B,  from  the  circumcentre,  the  incentre,  the  centroid,  and  the 
orthocentre. 

3.  In  the  ambiguous  triangle  show  that  the  circumcircles  of  the  two 
triangles,  when  there  are  two  solutions,  are  equal. 

4.  Prove  that  1  +  1  +  1  =  1. 

ra      rb      rc      r 

5.  In  any  triangle  prove  A  =  Vr  rarbrc. 

6.  Prove  that  the  product  of  the  distances  of  the  incentre  from  the 
vertices  of  the  triangle  is  4  r2R. 

7.  Prove  that  the  area  of  all  triangles  of  given  perimeter  that  can  be 
circumscribed  about  a  given  circle  is  constant. 

8.  Prove  that  the  area  of  the  triangle  ABC  is  .Rr(sin  A  +  sin  B -j- sin  C). 


CHAPTER   IX. 

• 

SERIES  —  DE   MOIVRE'S   THEOREM  —  HYPERBOLIC 
FUNCTIONS. 

77.  First  consider  some  series  by  means  of  which  loga- 
rithms of  numbers  and  the  natural  functions  of  angles  may 
be  computed.  For  this  purpose  the  following  series  is 
important  : 


It  may  be  derived  as  follows  : 
By  the  binomial  theorem, 


=  1  |  nx  -      I.-  \ 


[2  n2  [3 


n  n  n 


I 
xx  --     *  -- 

n    ,      \        n/\       n 

+ 


and  if  w  increase  without  limit, 


This  is  called  the  exponential  series,  and  is  represented  by 
ex,  so  that 


It  is  shown  in  higher  algebra  that  this  equation  holds  for 
all  values  of  x  ;  whence,  if  x  =  1, 


97 


98  PLANE   TRIGONOMETRY. 

This  value  of  e  is  taken  as  the  base  of  the  natural  or 
Naperian  system  of  logarithms. 

This  value  c,  however,  is  not  the  base  of  the  system  of  logarithms 
computed  by  Napier,  but  its  reciprocal  instead.  The  natural  logarithm 
is  used  in  the  theoretical  treatment  of  logarithms,  and,  as  will  presently 
appear,  it  is  customary  to  compute  the  common  logarithm  by  first 
finding  the  natural,  and  then  multiplying  it  by  a  constant  multiplier 
called  the  modulus,  Art.  82 ;  i.e.  in  the  Naperian  system  the  modulus 
is  taken  as  1,  and  the  base  is  computed.  In  the  common  system  the 
base  10  is  chosen  and  the  modulus  computed. 

78.  From  the  exponential  series  the  value  of  e  may  be 
computed  to  any  required  degree  of  accuracy. 


^=2-5 

1  =  0.1666666666 
l§ 

1  =  0.0416666666 

li 

1  =  0.0083333333 
\2. 

1=0.0013888888 
1  =  0.0001984126 

1=0.0000248015 
15 

1  =  0.0000027557 

[9 

—  =0.0000002755 


Adding,  e  =  2.7182818,  correct  to  7  decimal  places. 


SERIES.  99 

79.  To  expand  a*  in  ascending  powers  of  x. 


Let  ax  =  ez,  then  z  =  loge  ax  =  x  •  loge  a.          (Arts.  35,  40) 
Substituting 

„*=!  +  *.  log,  «  +  ^°f  a>2  +  ^<f  ")3  +  .... 

l±  IS 

Now  put  1  +  a  for  a,  and 


a*  [log.  (1  +  a)]1  . 


[2  18 

But  by  the  binomial  theorem, 


L±  12 

Equating  coefficients  of  x  in  the  second  members  of  the 
above  equations, 

i       ^  a2  ,  a3      #4  , 

log,  (!  +  «)=  a-  _  +  _-_  +  ...; 

or  writing  a?  for  a, 


In  this  form  the  series  is  of  little  practical  use,  since  it 
converges  very  slowly,  and  only  when  x  is  between  +  1  and 
—  1  (higher  algebra). 

Put  —  x  for  #,  and 


.-.    log.  (1+Z)-  log,  (1- 


100 


PLANE   TRIGONOMETRY. 


Finally,  put  -  -  -  for  x,  and 

2i  n  -j-  1 


=  log, 


-  log,  n 


, 


I 


a  series  which  is  rapidly  convergent. 

80.  From  this  series  a  table  of  logarithms  to  the  base  e 
may  be  computed. 

To  find  loge  2  put  n  =  1.  Then,  since  loge  1  =  0,  the  series 
becomes 


3     3  •  33     5  •  35     7  •  37     9  •  39 


The  computations  may  be  arranged  thus  : 


2.00000000 
.66666667 


=.66666667 


.  07407407  H-  3  =  .02469136 
.00823045-  5  =  .00164609 
.00091449-  7  =  .00013064 
.  00010161  -f-  9  =  .00001129 
.00001129  -  11  =  .00000103 
.00000125  -  13  =  .00000009 


.69314717 

whence  loge  2  =  0.693147,  correct  to  6  decimal  places. 
To  find  loge  3>  Put  n  =  2,  and 
log  3  =  loge2  + 


5 

25 
25 
25 
25 


SERIES.  101 

2.00000000 

.40000000    =  .40000000 

.01600000-^3  =  .00533333 

.00640000-5=  .00012800 

.00025600-^-7  =  .00000366 

.00000102 -9=  .00000011 


.40546510 
log,  2=  .69314717 

...  log,  3  =  1.098612, 
correct  to  6  decimal  places. 

log,  4  =  2  x  log,  2,  log,  6  =  log,  3  +  log,  2,  etc.  (Why  ?) 
The  logarithms  of  prime  numbers  may  be  computed  as  above 
by  giving  proper  values  to  n. 

81.  Having  computed  the  logarithms  of  numbers  to  base  e, 
the  logarithms  to  any  other  base  may  be  computed  by  means 
of  the  following  relation  : 

Let  l°g"a  n  =  x  >  then  ax  =  n. 

Also,  Iog6  n  =  y\  then  bv  =  w, 


Hence,  loga  (a*)  =  logrt  (ft*)  , 

and  .-.  x  =  ylogab. 

It  follows  that  loga  n  =  logb  n  -  loga  b  ; 

whence  Iog6  n  =  logffl  n 


This  factor  -  is  called  the  modulus  of  the  system  of 

loga  6 

logarithms  to  base  b.  Using  it  as  a  multiplier,  logarithms 
of  numbers  to  base  b  are  computed  at  once  from  the  loga- 
rithms of  the  same  numbers  to  any  other  base  a. 


102  PLANE   TRIGONOMETRY. 

82.  To  compute  the  common  logarithms. 

Common  logarithms  are  computed  from  the  Naperian  by 

use  of  the  modulus ;  i.e. 

log, 10 

Iog10  n  —  loge  n  • — 

By  Art.  80,  loge  10  can  be  found,  and 

=  .434294,  the  modulus  of  the  common  system. 

loge  10 

Ex.   Compute  the  common  logarithms  of : 

2,     3,     4,     6,     5,     10,     15,     216,     3375. 

COMPLEX  NUMBERS. 

83.  In  algebra  it  is  shown  that  the  general  expression  for 
complex  numbers  is  a  +  bi,  where  a  represents  all  the  real 
terms  of  the  expression,  b  the  coefficients  of  all  the  imagi- 
nary terms,  and  i  is  so  denned  that  i2  =  —  1 ;  whence 

i  =  V—  1,   i2  =  —  1,   i5  =  —  i,   ft  =  1,  etc. 

The  laws  of  operation  in  algebra  are  found  to  apply  to 
complex  numbers.  Moreover,,  -it  is  ftwther  shown  that  if 
two  complex  numbers  are  equal,  the  real  terms  are  equal, 
and  the  imaginary  terms  are  equal;  i.e.  if 

a  +  bi  =  c  +  di,     fe 
o^^  \ji  '  £ 

then  a  =  fi  and  -£=  d. 


x- 


Filially,  the  complex  number 
may  be  graphically  represented  as 
follows : 

•X  The  real  number  is  measured 
along  OX,  a  units ;  the  imaginary 
parallel  to  OY,  b  units.  The  line 

r   is   a   graphic    representation   of 
FIG.  43.  , . 

a  +  bi. 


DE  MOIVRE'S   THEOREM.  103 

Since  a  =  r  cos  6  and  b  =  r  sin  0, 

/.  a  +  bi  =  r  (cos  6  +  i  sin  0). 

The  properties  of  complex  numbers  are  best  developed  by 
using  this  trigonometric  form.  If  r  be  taken  as  unity,  then 
cos  6  +  i  sin  6  represents  any  complex  number. 

84.  De  Moivre's  Theorem.  To  prove  that,  for  any  value 
of  w, 

(cos  0  +  i  sin  8)n  =  cos  n0  -f  *  sin  nQ. 

I.  When  n  is  a  positive  integer. 
By  multiplication, 

(cos  a  +  i  sin  a)  (cos  /3  +  i  sin  y8) 

=  cos  a  cos  /3  —  sin  a  sin  £  +  *  (sin  a  cos  /3  +  cos  a  sin  /3) 
=  cos(a  +  /3)  +  i  sin  (a  +  £). 
In  like  manner, 
(cos  a  +  t  sin  a)  (cos  @  -{-  i  sin  /3)  (cos  7  +  i  sin  7) 

=  cos(«  +  £  +  7)  +  i  sin  («  +  P  +  7)  ; 
and  finally, 
(cos  a-\-i  sin  a)  (cos  /3  +  i  sin  /3)  (cos  7  +  ^  sin  7)  •••to  %  factors 

=  cos(«  +  13  +  7  +  •••)  +  i  sin(<*  +  £  +  7  4-  — )• 
Now  let  a  =  /9  =  7  =  •••,  and  the  above  becomes 
(cos  a  +  i  sin  «)w  =  cos  na  -f- 1  sin  wa. 

II.  When  w  is  a  negative  integer. 

\  fW 

Let  w  =  —  m ;  then 

<-    W         'V^^*-/VvN^^     .    tY*C 

(cos  a  +  i  sin  a)n  =  (cos  a  +  i  sin  a)~m  ^  ^a-vY^" 

1 1 

(cos  a  +  i  sin  a)TO      cos  ma.  +  i  sin  ma 
cos  Twa  —  i  sin  ma 


(cos  wa  +  i  sin  m«)(cos  ^a  —  i  sin  ma) 

cos  met  —  j  sin  m« 

cos2  ma  +  sin2  ma 

cos  ma  —  i  sin  ma  =  cos  (  —  m)  a  +  i  sin  (  —  m)  a. 


104  PLANE   TRIGONOMETRY. 

Substituting  n  for  —  m,  the  equation  becomes 
(cos  a  -f-  i  sin  a)"  =  cos  no.  +  i  sin  na. 

III.    When  n  is  a  fraction,  positive  or  negative. 

P 
Let  n  =  — ,  JP  and  ^  being  any  integers. 

Now 


a       .    .     «V  a       .    .          a  ...          xl      TN 

cos  -4^  sm  -     =  cos  a  -  -  +  ^  sin  0  •  -  =  cos  a  +  i  sin  a   (Thy  I). 
q  qj  q  q 


Then 


cos  -  -M  sin  -  )  =  (cos  a  +  i  sin  a 


Raising  each  member  to  the  power  p, 

P      (        a  a\p  p  p 

(cos  a  -f  ^  sm  «)«  =    cos  -  + 1 8m  -  I  =  cos  —  «  +  i  sin  —  a. 

\     q          qJ         q  q 


COMPUTATIONS   OF  NATURAL  FUNCTIONS. 

85.  The  radian  measure  of  an 
acute  angle  is  greater  than  its  sine 
and  less  than  its  tangent,  i.e. 

sin  a<a<  tan  «. 

Let  «  be  the  circular,  or  radian, 
measure  of  any  acute  angle  AOP. 
Then,  in  the  figure, 

area  of  sector  OAP  <  area  of  triangle  OAT, 
i.e.   i  OA  •  arc  AP  <  %  OA  •  AT. 


Now,  since 


NP  <  arc  AP, 

NP     arcAP     AT 
OP         OP     "  OP' 


But 

whence 


arc  AP 
OP 


=  circular  measure  of  AOP  =  «  ; 
sin  a  <  a  <  tan  a. 


NATURAL  FUNCTIONS.  105 

86.    Since  sin  a  <  a  <  tan  a, 


sin  a      cos  a 

Hence,  however  small  «  may  be,  lies  between  1  and 

_,  sin « 

.      When    a    approaches    0,    cos  a    approaches     unity. 

cos  a 

Therefore,    by    diminishing   a   sufficiently,   we    may    make 

- — -  differ  from  unity  by  an  amount  less  than  any  assign- 
sin  a 

able  quantity.     This  we   express   by  saying  that  when    « 

approaches  0, approaches  unity  as  a  limit,  i.e.  =  1, 

sin  a  sin  « 

approximately.     Multiplying  by  cosa(=  1,  nearly),  we  have 

=  1,    approximately.      Whence,    if    a    approaches    0, 
tan  a 

tan  a  =  sin  a  =  a,  approximately. 

87.   Sine  and  cosine  series, 
cos  no.  -f-  i  sin  ncc  =  (cos  a  +  i  sin  «)",  (De  Moivre's  Theorem). 

Expanding  the  second  member  by  the  binomial  formula, 
it  becomes, 

n     1  •  i     W  \n  1  )  „     o  .9.9 

cos  «  +  n  cos      «  •  %  sin  «  H ^ —   — -  cos""  a  .  ^J  snr* « 


,  n(n  —  l}(n  —  2)       _  o        .o   .   , 
H  --  i—  —  ^  cos""3  a  •  ^3  sm3  a 


Substituting  the  values  of  z2,  z3,  ^4,  etc.,  we  have 

cos  na  +  i  sin  w«  =  cos"  «  —  -^-cos""2  a  sin2  a 

[2 

n(w-l)(n-2)fw-3) 
H  —  ^  -  "•  —  —  i-  cos""4  «  sin4  a  —  ... 


n  cos""1  a  sin  «  -         ~  ~      cos"-3  «  sin3 

[3 


106  PLANE   TRIGONOMETRY. 


Equating  the  real  and  imaginary  parts  in  the  two  members, 

n(n—V)      „  9      .  o 
cos  na  =  cosn  a  --  ^  —  -  -  ^  cosn~~  a  sin  a 


, 

and     «m  no,  = 


Ex.  1.     Find   cos  3  a  ;   sin  3  a. 

In  the  above  put  n  =  3,   and   cos  3  a  =  cos3  «  —  3  cos  a  sin2  a 

=  4  cos3  a  —  3  cos  a  ; 
also  sin3  a  =  3  cos2  a  sin  a  —  sin3  a 

=  3  sin  a  —  4  sin3  «. 

2.   Find  sin  4  a  ;   cos  4  a  ;   sin  5  ft  ;   cos  5  a. 

It  will  be  noticed  that  in  the  series  for  cos  na  and  sin  na  the  terms 
are  alternately  positive  and  negative,  and  that  the  series  continues  till 
there  is  a  zero  factor  in  the  numerator. 

88.   If  now  in  the  above  series  we  let  na  =  6,  then 


„  o      •   o 
cos  6  =  cos"  «  --  rj=  -  cos""2  a  sin2  a 

I* 


<A  /sin« 

=  cosn«  —  ^-   —  ^cosw~2a[- 

V    «   / 


cos"~*  a  sin 


4      sn  « 

" 


If  now  0  remain  constant,  and  a  decrease  without  limit, 
then  will  n  become  indefinitely  great,  and  -      -  and  every 


NATURAL  FUNCTIONS.  107 

power   thereof,  and   cos  a   and   every   power   of   cos  a  will 
approach  unity  as  a  limit,  so  that 

e*    e*_e6 

'[2     [4     [6 
Similarly,  sin  0  =  8  -  f-  +  f~  -  ~  + . . .. 

[o      |o      I  ? 

By  algebra  it  is  shown  that  these  series  are  convergent  for 
all  values  of  0.  By  their  use  we  can  compute  values  of  sin  0 
and  cos  6  to  any  required  degree  of  accuracy. 

$3      2  $5 

Show  from  the  above  that  tan  6  =  0  +  — -  +  ~^-=-  +  •••• 

o        lo 

Ex.  1.   Compute  the  value  of  sin  1°,  correct  to  5  places. 

In  sin0  =  0-  — +  —-  —  +  .»,   make  0    the   radian 

[3      |5      [7 

measure  of  1°  =  -£-  =  0.01745  + . 

180 

Then,  0  =  0.01745  + 

£=0.0000008. 

[£ 

.-.  sin  0  =  0.01 745+. 

The  terms  of  the  series  after  the  first  do  not  affect  the  fifth  place,  so 
that  the  value  is  given  by  the  first  term,  an  illustration  of  the  fact  that, 
if  a  is  small,  sin  a  =  a,  approximately.  Compare  the  value  of  tan  1°. 

2.  Show  that  sin  10°  =  0.17365 ;  c6s  10°  =  0.98481  •;  sin  15°  =  0.25882 ; 
cos  60°  =  0.50000. 

3.  Find  the  sine  and  cosine  of  18°  30' ;  22°  15' ;  67°  45'. 

It  is  unnecessary  to  compute  the  functions  beyond  30°,  for  since 

sin  (30°  +  0)  +  sin  (30°  -  0)  =  cos  0  (why?), 
.-.  sin  (30°  +  0)  =  cos  9  -  sin  (30°  -  0). 
So,  also,  cos  (30°  +  0)  =  cos  (30°  -  0)  -  sin  0. 

Giving  0  proper  values  the  functions  of  any  angle  from  30°  to  45°  are 
determined  at  once  from  the  functions  of  angles  less  than  30°. 

Thus,  sin  31°  =  cos  1°  -  sin  29° ; 

cos  31°  =  cos  29°- sin  1°. 

4.  Find  sine  and  cosine  of  40° ;  of  50°. 


108  PLANE  TRIGONOMETRY. 

89.   The    following    are    sometimes    useful    in    applied 
mathematics : 

Ex.  1.   To  find  the  sum  of  a  series  of  sines  of  angles  in  A.  P.,  such  as 
sin  a  +  sin  (a  +  /?)  +  sin  (a  +  2  /?)  +  ...  +  sin  («  +  [  n  -  I]/?). 

2sinasin£:=  cosf  a  —  ^-j  —  cos  la  +^)> 

/3  /  /?\  /  *^  /«?\ 

2  sin  (a  +  /?)  sin^  =  cos  f  a  +  "  )  —  cos  (  a  +  -^  J, 

2 sin  (a  +  2/?)siii^  =  cos(  a  +  — p )  -cos  (a  +—^), 
\  2  /  \          J   / 


2  sin  (a  +  [n  -  1]0)  sin     =  cos    a  +      ~-         -  cos    a  + 

Adding 
2  {sin  a  +  sin  («  +  £)+  sin  (a  +  20)+-+  sin  (a  +  [n  -  1]  0)}  sin  | 


.-.  sin  ce  +  sin  (a  +  0)  +  sin  (a  +  2  /?)  +  ..,  +  sin  (a  +  [n  - 


sin 


Similarly  it  can  be  shown  that 

cos  a  +  cos  («  +  0)  +  cos  (a  +  2  0)  +  —  +  cos  (a  +  [n  -  I]/?) 


.    fl 

8111  9 


HYPERBOLIC  FUNCTIONS.  109 

90.    The  series  ex  =  1  +  x  +  ^  +  -  +  —  +  p  +  —  is  proved 

l±    l£         ii 

in  higher  algebra  to  be  true  for  all  values  of  a?,  real  or 
imaginary.     Then  if  a:  =  i'0, 


(<^ ei0  =  cos  0  +  i  sin  0  (Art.  87). 
In  like  manner,          e~ie  =  cos  0  —  i  sin  0. 
Adding,  cos  0  = ; 

subtracting,  sin  0  =  -    — 


HYPERBOLIC   FUNCTIONS. 

giO  _   g-iO  _ 

91.   Since    sin  0  =  —  •£-.  -  ,    and    cos  0  =  --  —        are 
true  for  all  values  of  0,  let  6  =  id. 

Then,      sin  (16)  =  e~'~  e&       e°  ~e~ 


'° 


and  cos  (iO)  =  =  cosh  6, 

f  ./jx      sin  (id)      i  sinh  0       .  ,      ,   /j 
so  that        tan  (i0)  =  --  ^-—~  =  -     —  r  =  ^  tanh  0, 
cos  (*0)       cosh  0 


where  smA  0,  eos^  0,  ^9iA  ^,  are  called  the  "hyperbolic  sine, 
cosine,  and  tangent  of  0.  The  hyperbolic  cotangent,  secant, 
and  cosecant  of  0  are  obtained  from  the  hyperbolic  sine, 
cosine,  and  tangent,  just  as  the  corresponding  circular  func- 
tions, cotangent,  secant,  and  cosecant,  are  obtained  from 
tangent,  cosine,  and  sine.  The  hyperbolic  functions  have 
the  same  geometric  relations  to  the  rectangular  hyper- 


110  PLANE   TRIGONOMETRY. 

bola  that  the  circular  functions  have  to  the  circle,  hence  the 
name  hyperbolic  functions. 

sinh  8  =  — — ^ — ,        .  •.  csch  6  =  — 5 


tanhe  =  21^  L_,        ...  CothO  =  ^-±  2-j. 

92.  From  the  relations  of  Art.  91  it  appears  that  to  any 
relation  between  the  circular  functions  there  corresponds  a 
relation  between  the  hyperbolic  functions. 

Since  cos2  (id)  +  sin2  (16)  =  1, 

cosh2  0  +  i2  sinh2  0  =  1, 
or  cosh2  0  —  sinh2  0  =  1. 

This  may  also  be  derived  thus: 

cosh2  0- sinh2  0=/^±-^ 
V      * 


Also  since 

sin  (i«  +  z/8)  =  sin  (i«)  cos  (1/8)  +  cos  (i«)  sin  (e/8), 

/.  i  sinh  (a  +  /3)  =  i  sinh  a  cosh  /3  +  cosh  a  •  i  sinh  & 
and        sinh  (a  +  /8)  =  sinh  a  cosh  /3  +  cosh  a  sinh  /8. 

Let  the  student  verify  this  relation  from  the  exponential 
values  of  sinh  and  cosh. 

EXAMPLES. 
Prove 

1.  cosh  (a  +  /?)  =  cosh  a  cosh  ft  +  sinh  a  sinh  J3. 

2.  cosh  (  «+  (3)  -  cosh  (a  -  ft)  =  2  sinh  «  sinh  /?. 

3.  cosh  20  =  1  +  2  sinh2  6  =  2  cosh2  0  -  1. 

4.  sinh  2  a  =  2  sinh  a  cosh  a. 


EXAMPLES.  HI 


h^-    /1+cosn0.     inh--    /cosh  0  ~  1 

6.  sinh  30  =  3  sinh  0  +  4  sinh3  0. 

7.  sinh  0  +  sinh  <£=  2  sinh— 1-2 cosh  —  p~2. 

8.  sinh  a  +  sinh  («  +  /?)+  sinh  («  +  2  y8)  +  ...+  sinh  («  +  [n  - 

sinh  f  a  +  n  ~~ 


Binhl 

4-r. nV,  /3    i    ^ 

9. 


10.  sinh-1  x  —  cosh"1  Vl  -f  a;2  =  tanh-1 — . 

Vl  +  a;2 

11.  cosh  (a  +  /?)  cosh  (a  -  /?)  =cosh2  a  +  sinh2  /?  =  cosh2  /?  +  sinh2  a. 

12.  2  cosh  n«  cosh  a  =  cosh  (n  +  1)  a  +  cosh  (n  —  1)  a. 

13.  cosh  a  =  £(e«  +  «~a)  =  1  +  —  +  —  +  ••-. 

14.  sinh  a  =  \  (e»  -  e-a)  =  a  +  ^  +  ^  +  .... 

15   15 

15.  tanh-1  a  +  tanh-1  b  =  tanh-1  -^-±1. 


SPHERICAL   TRIGONOMETRY. 

CHAPTER  X. 

SPHERICAL    TRIANGLES. 

93.  Spherical  trigonometry  is  concerned  chiefly  with  the 
solution  of  spherical  triangles.  Its  applications  are  for  the 
most  part  in  geodesy  and  astronomy. 

The  following  definitions  and  theorems  of  geometry  are 
for  convenience  of  reference  stated  here. 

A  great  circle  is  a  plane  section  of  a  sphere  passing 
through  the  centre.  Other  plane  sections  are  small  circles. 

The  shortest  distance  between  two  points  on  a  sphere  is 
measured  on  the  arc  of  a  great  circle,  less  than  180°,  which 
joins  them. 

A  spherical  triangle  is  any  portion  of  the  surface  of  a 
sphere  bounded  by  three  arcs  of  great  circles.  We  shall 
consider  only  triangles  whose  sides  are  arcs  not  greater  than 
180°  in  length. 

The  polar  triangle  of  any  spherical  triangle  is  the  triangle 
whose  sides  are  drawn  with  the  vertices  of  the  first  triangle 
as  poles.  If  ABO  is  the  polar  of  A'B'C',  then  A'B'O'  is  the 
polar  of  ABC. 

In  any  spherical  triangle, 

The  sum  of  two  sides  >  the  third  side. 

The  greatest  side  is  opposite  the  greatest  angle,  and  conversely. 
Each  angle  <   180°  ;    the  sum  of  the  angles  >  180°,  and 
<   540°. 

Each  side  <  180°  ;  the  sum  of  the  sides  <  360°. 

112 


SPHERICAL  TRIANGLES. 


113 


The  sides  of  a  spherical  triangle  are  the  supplements  of  the 
angles  opposite  in  the  polar  triangle,  and  conversely. 

If  two  angles  are  equal  the  sides  opposite  are  equal,  and 
conversely. 

The  sides  of  a  spherical  triangle  subtend  angles  at  the 
centre  of  the  sphere  which  contain  the  same  number  of  angle 
degrees  as  the  arc  does  of  arc  degrees;  i.e.  an  angle  at  the 
centre  and  its  arc  have  the  same  measure  numerically. 

The  arc  does  not  measure  the  angle  for  they  have  not  the  same  unit 
of  measurement,  but  we  say  they  have  the  same  numerical  measure  ;  i.e. 
the  arc  contains  the  unit  arc  as  many  times  as  the  angle  contains  the 
unit  angle. 

The  angles  of  a  spherical  triangle  are  said  to  be  measured 
by  the  plane  angle  included  by  tangents  to  the  sides  of  the 
angle  at  their  intersection.     They  have  therefore  the  same 
numerical  measure  as  the  dihe- 
dral angle  between  the  planes 
of  the  arcs. 

In   the  figure   the  following 
have  the  same  numerical  meas-   * 
ure  : 

arc  a  and  angle  a ; 

arc  b  and  angle  y3 ; 

arc  c  and  angle  7 ; 

plane  angle  A1 BO'-, 

spherical  angle  B  and  dihedral  angle  A-BO-C; 

spherical  angle  C  and  dihedral  angle  B-C 0-A; 

spherical  angle  A  and  dihedral  angle  C-A  0-B. 

A'C'B  and  C'A'B  have  not  the  same  measure  as  spherical^  angles 
C  and  A,  for  BA',  A'C',  C'B  are  not  perpendicular  to  OA  or  OC. 

94.  In  plane  trigonometry  the  trigonometric  functions 
were  treated  as  functions  of  the  angles.  But  since  an  angle 
and  its  subtending  arc  vary  together  and  have  the  same 


114 


SPHERICAL   TRIGONOMETRY. 


numerical  measure,  it  is  clear  that  the  trigonometric  ratios 
are  functions  of  the  arcs,  and  may  be  so  considered.     All 
the  relations  between  the  functions  are  the  same  whether  we 
consider  them  with  reference  to  the  angle 
or  the  arc,  so  that  all  the  identities   of 
plane  trigonometry  are  true  for  the  func- 
a    tions  of  the  arcs. 
I         Thus  in  the  figure  we  may  write, 


x 
FIG.  46. 


y 


y 


sm  a  =  -  or  sin  a  =  -  ; 


sin2  a  -f-  cos2  a  =  1,   or   sin2  a  +  cos2  a  =  1 ; 
cos  2  a  =  2  cos2  a  —  1,    or    cos  2  a  —  2  cos2  a  —  1. 

GENERAL   FORMULAE   FOR  SPHERICAL   TRIANGLES. 

95.    The  solutions  of  spherical  triangles  may  be  effected 
by  formulae  now  to  be  developed: 

First  it  will  be  shown  that  in  any  spherical  triangle 

cos  a  =  cos  £cos  c  +  sin  b  sin  e  cos  A, 

<>         U.  [^         U( 

cos  b  =  cos  c  cos  a  +  sin  c  sin  a  cos  B, 
cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C. 

The  following  cases  must  be  considered  : 
I.    Both  b  and  c <  90°.  III.    Both  b  and  c>  90°. 

II.    b  >  90°,  c  <  90°.  IV.    Either  b  or  c  =  90°. 

V.    6  =  <?  =  90°. 

The  figure  applies  to  Case  I. 

Let  ABC  be  a  spherical  tri- 
angle, a,  6,  c  its  sides,  and  (9 
the  centre  of  the  sphere. 

Draw  AC'  and  AB'  tangent 
to  the  sides  5,  c  at  A.  (The 
same  result  would  be  obtained 
by  drawing  AB' ,  AC'  perpen- 
dicular to  OA  at  any  point  to 


GENERAL  FORMULAE.  115 

meet  OB,  00.)  Since  these  tangents  lie  in  the  planes  of 
the  circles  to  which  they  are  drawn,  they  will  meet  00  and 
OB  in  0'  and  B' ,  and  the  angle  C1  AB'  will  be  the  measure 
of  the  angle  A  of  the  spherical  triangle  ABC.  Since  OAB' , 
OAOf  are  right  angles,  A  OB',  A  00'  must  be  acute,  and 
hence  sides  <?,  b  are  each  <  90°. 
In  the  triangles  C' AB'  and  C'OB', 

C'B'2  =  AC'2  +  AB'2  -  2  AC'  -  AB'  cos  C'AB', 
and       B' 0'*  =00'*+  OB'2  -200'  -  OB'  cos  C'OB'. 
Subtracting  and  noting  that 

cos  C' AB'  =  cos  A  and  cos  C'OB'  =  cos  a, 
we  have 

0  =  OO'2  -  A  C'2  +  OB'2  -  AB'2 

+  2  AC'  -  AB'  cos  A  -  2  00'  .  OB'  cos  a. 
But     OO'2  -  A  C'2  =  OJ2  and  O#'2  -  AB'2  =  OA2. 
Hence,         0  =  OA2  +  A  C"  -  AB'  cos  -A-OC'.  OB'  cos  a ; 
0J.      OA       AC'     AB' 

cosa  = 


.*.  cos  a  =  cos  b  cos  c  -f-  sin  b  sin  c  cos  A. 
Similarly, 

cos  b  =  cos  a  cos  c  -f  sin  a  sin  c  cos  .#, 
and  cos  c  =  cos  a  cos  6  4-  sin  a  sin  &  cos  O. 

These  formulae  are   important,   and   should   be  carefully 
memorized. 

II.    6>90°;    <7<90°.  A<- 

In  the  triangle  ABC,  let  6  >90° 
and  £<90°.     Complete   the   lime  FIG.  48." 

BAG  A'.      Then    in    the    triangle 
A'CB  the  sides  a  and  A'C  are  both  less  than  90°,  and  by  (I) 

cos  A'B  =  cos  A'C  cos  a  +  sin  A'<7  sin  a  cos  A'CB. 


116  SPHERICAL   TRIGONOMETRY. 

But  A'B  =  lSQ°-c,  A'C=lSQ°-b,  and  A CB =~LSQ°  -  C. 

.*.  cos  (180°  -  <?)  =  cos  (180°  -  5)  cos  a 

+  sin  (180°  -  5)  sin  a  cos  (180°  -  C) ; 

or         —  cos  c  =  (—  cos  6)  cos  a  -f-  sin  6  sin  a  (—  cos  (7)? 
and  cos  c  =  cos  a  cos  6  -f-  sin  a  sin  6  cos  C. 

A  similar  proof  will  apply  in  case  c  >  90°,  b  <  90°. 

III.    Both  b  and  <?  >  90°. 

the  triangle  ABC,  let  both 
arid    <?>90°.      Complete    the 


FlG-  49' 


lune  ABA'C.      Then  since 
and  A'B  are  both  <  90°, 

cos  A'B  +  sin  J/<7  sin  A'B  cos  J/. 


cos  a  =  cos 
But    A'  =  A, 

.-.  cos  a  =  cos  (180°  -  5)  cos  (180°  -  c) 

+  sin  (180°  -  6)  sin  (180°  -  c)  cos  A  ; 

or        cos  a  =  cos  6  cos  c  -f  sin  6  sin  c  cos  .A; 

Cases  IV  and  V  are  left  to  the  student  as  exercises. 

96.    Since  the  angles  of  the  polar  triangle  are  the  supple- 
ments of  the  sides  opposite  in  the 
first  triangle,  we  have 

a'  =  180°  -A,    V  =  180°  -B, 
c'=1800-<7,    ^'  =  180°  -a. 

Substituting  in 

cos  a'  =  cos  b'  cos  c' 

-f-  sin  bf  sin  c'  cos  A 
we  have 

-  cos  (180°  -A)=  cos  (180°  -  B)  cos  (180°  -  C) 

+  sin  (180°  -  B)  sin  (180°  -  (7)  cos  (180°  -  a); 

or       —  cos  A=    —  cos  -B)(  —  cos  <7)  +  sin  B  sin  <7(  —  cos  a). 


GENERAL  FORMULA.  117 

Changing  signs, 

cos  A  -  -  cos  B  cos  C  +  sin  B  sin  C  cos  a. 

(J         ;  <J      i/i/ 

Similarly,       cos  B  —  —  cos  A  cos  C  +  sin  A  sin  C  cos  6, 

I//  U       \S          U 

and  cos  C  =  -  cos  ^1  cos  .B  +  sin  A  sin  B  cos  c. 

,      .     ,  ,  .       -,  sin  A     sin  B     sin  C 

97.   In  any  spherical  triangle  to  prove  -     —  =—    —  =  —  — 

sin  a      sin  b      sin  c? 

cos  a  —  cos  6  cos  c 


Since  cos  J.  = 

.*.  sin2  A  =1  — 


sin  b  sin  < 
cos  a  —  cos  b  cos 


sin  5  sin  c 

_  sin2  6  sin2  c  —  (cos  a  —  cos  b  cos  g)2 
sin2  b  sin2  c 

_  (1  —  cos2  5)  (1  —  cos2  c)  —  (cos  a  —  cos  6  cos  g)2 
sin2  6  sin2  c 

1  —  cos2  «  —  cos2  b  —  cos2  c  +  2  cos  a  cos  6  cos  c 
sin2  6  sin2  <? 

Hence, 

Vl  —  cos2  a  —  cos2  b  —  cos2  c  —  2  cos  a  cos  5  cos  c 
sm  J.  =  —  — : — - — : — 

sm  6  sm  c 

sin  ^1  _  Vl  —  cos2  a  —  cos2  b  —  cos2  c  —  2  cos  a  cos  6  cos"g 
sin  a  sin  a  sin  6  sin  c 


By  a  similar  process,  sm       and  -7- —  will  be  found  equal 
sin  b  sin  c 

to  the  same  expression. 

.   sin  ^t  _.  sin  B  _  sin  Ct 
sin  a      sin  6      sine 


118  SPHERICAL   TRIGONOMETRY. 

98.    Expressions  for  sine,  cosine,  and  tangent  of  half  an 
angle  in  terms  of  functions  of  the  sides. 

A 

We  have    2  sin2  —  =  1  —  cos  A 
2t 

_  -,       cos  a  —  cos  b  cos  c 
sin  b  sin  c 

_  cos  b  cos  c  +  sin  b  sin  c  —  cos  a 
sin  b  sin  c 

_  cos  (6  —  <?)  —  cos  a 
sin  6  sin  £ 

Then    2  sin^  =  2  sin  Ka  +  ft~  e>  «in  Ka  -  b  +  0  (Art.  51) 

2  sin  5  sin  <? 


2  sin  (s  —  6)  sin  (  s  —  c) 

-  i  -  ^  -  —  :i_ 

sin  6  sin  c 


when 


A  _^  /sin  (8  -  b}  sin  (a  -  c) 
-       sin 6  sine 


Similarly,  sin  ^  = 

J 


sin  a  sin  c 

and  sin  C       Sto£=b)tin (s  -  a\ 

2      N          sin  a  sin  ft 

Also  from  the  relation 

2  cos2  —  =  1  +  cos  A 

_  -j       cos  a  —  cos  b  cos  c 
sin  b  sin  c 


we  have  cos  f  =  Jsins  sin(f  -  «) 

2       *      sin  ft  sm  c 


Also,  cos  ^r  = 


s  sin  (s  ~ 


2  sin  c  sin  a 


cos  ^  =\"""r  "~"\"  ^* 

2      x     sin  a  sm  ft 


GENERAL  FORMULAE.  119 


From  the  above,  A 


A_        2        /sin  (8 -ft)  sin  (8 -c) 
tan  2  ~     a  A      *      sin  s  sin(s  -  a) 


Also  tan  B  _    /sin  (s  -  a]  sin  (8  -  c) . 

A1S0'  n2~>      sins  sin  («  -ft) 


and  ton 

2      x      sin  s  sin  (s  -  c) 

Compare  the  formulae  thus  far  derived  with  the  corresponding  for- 
mulae for  solving  plane  triangles.  The  similarity  in  forms  will  assist 
in  memorizing  the  formulae  for  solving  spherical  triangles. 


99.   From  the  formulse  of  Art.  96,  the  student  can  easily 
prove  the  following  relations  : 


.    a  _  ^  /-cos£  cos(£  -  A") 
ln  2  ~^        sin  B  sin  C 


where 


•In     =  l. 


^  «  _  .  /cos  (S  -  JS)  cos  (8  -  C) 
>S2~>  sin^sinC 


tan  —  =  A  /     -  cos.S  cos(<Sf- 
2 


tw|=l, 


120  SPHERICAL   TRIGONOMETRY. 

100.   Napier's  Analogies. 


tan  —      -Jsin 

2        *      sin  s  sin  (s  —  a 
Since  =  v 

tan—      Jsin(g-  c)  ship -a) 
2        *       sin  s  sin  (s  —  b) 


_    /sin2(s  —  b)  _  sin  (s  —  b)  f 
^  sin2  (s  —  a)      sin  (s  —  a)  ' 


by  composition  and  division, 

tan  4  +  tan  ^ 

2  2  _  sin  (s  —  5)  +  sin  (s  —  a) 

J.  J9  ~"  sin  (s  —  6)  —  sin  (s  —  a)' 

tan  --  tan  — 

'-  — 


sin          sin 


. 


A  B 

i/~kQ  _      r*o^     - 
_2  _  2  =  sin  1(2  g  -  q  -  6)cos  ^(q  -  ft) 

A  B  ~  cos  i  (2  s  -  a  -  b)  sin  ^  (a  -  b)' 

¥  (Art.  51) 


cos  —      cos  — 
2  2 


sin  ^(A  +  -g)  =  tan  1(2  g  -  a  -  b) 
sin  i(J.  -  -B)  "       tan  J(a  -  6) 


tan- 

-TJ— JT-,    since  2s  —  a  —  b  =  c. 


- 
.-.tan  §  (a-  6)=-  -tan|. 


To  find  an  expression  for  tanJ(J.  -  B)  we  have  only  to 
consider  the  polar  triangle,  and  by  substituting  180°  —  A  for 
a,  etc.,  180°  —  a  for  A,  etc.,  we  have  the  following  relations  : 

-  A  ~  180°  +  #  = 


GENERAL   FORMULA.  121 

also,   J(4-J9)=-£O_&); 

-  «  +  180°  -  *  = 


and  |=90°-f 


The  formula  then  becomes,  applying  Art.  29, 

sin  3  (a  -  &)       ^ 

ten  1  (.1  -*)=_«_       _cot|. 

sin    (a  +6) 


Formulae  for  tan^(a  +  6),  tan  %(A  -f-  5)  are  derived  as 
follows  : 

Since 


tan—  -  tan"= 


2  2      ^  sin  *  •  sin  (s  —  a)        *     sin  8  •  sin  (s  —  6) 


sin  —  sin  — 
2        2 


—        2 

By  composition  and  division, 

A      B  ,       A  •   B 

cos— cos—  -f-  sin— sin  — 

2        2  2        2  =  sin  s  +  sin  (s  -  c) 

cos|cos|  -  sin|sin|  ~  sin  s  ~  sin  («  ~  ^ 

cos  i  (A  —  J5 )      tan  i  (a  +  J) 
whence  '-  = •_: — ! — L,          f  Art.  51) 

-KA  +  .B)       tan| 

since  2s  —  c=a  +  5, 


- 
or,  ten  |  (a +  6)  =  -  -tan|. 


122 


SPHERICAL   TRIGONOMETRY. 


The  value  of  tan  J  (A  -+-  J9)  is  derived  by  substituting  in 
terms  of  the  corresponding  elements  of  the  polar  triangle. 


.-.  tan 


cos|(a-6)       c 

* COt  -^. 


Similar  relations  among  the  other  elements  of  the  triangle 
may  be  derived,  or  they  may  be  written  from  the  above  by 
proper  changes  of  A,  B,  (7,  a,  b,  c  in  the  formulae.  The  stu- 
dent should  write  them  out  as  exercises. 

101.  Delambre's  Analogies. 

Since          sin  J  ( A  +  B)  =  sin  —  cos  —  4-  cos  —  sin  — , 

M  Jt  a  -a 

then  

sinK^  +  *)  =  Si"(*~*)+Sin(g~aMSing'Sin(87C)- 

sin  c  sin  a  •  sin  b 

(Art,  98) 


_H.ence« 


sn 


+ 


cos- 


sin  (g  —  b  )  +  sin  (s  —  a 
; 
sin  c 


2  sin  £  cos  |-  (a  —  6) 

2 ,     (Art.  51) 


^    •    c        c 
2  sin  -  cos  - 


and  Bto|(^  + 

In  like  manner  derive 


cos    (a  -  6) 

-      —  - 

coss 

B 


sin 


RIGHT  SPHERICAL  TRIANGLES.  123 


cos 


These  formulae  are  often  called  Gauss's  Formulae,  but  they  were  first 
discovered  by  Delambre  in  1807.  Afterwards  Gauss,  independently,  dis- 
covered them,  and  published  them  in  his  Theoria  Motus. 

102.  Formulae  for  solving  right  spherical  triangles  are 
derived  from  the  foregoing  by  putting  0=  90°,  whence 
sin  (7=1,  cos  (7=0. 

cos  c  =  cos  a  cos  b  +  sin  a  sin  6  cos  (7      (Art.  95) 
becomes  cos  c  =  cos  a  cos  ft.  (1) 

Substituting  the  value  of  cos  a  from  (1),  and  simplifying, 

cos  A  =  cos  «.  ~  cos  &  cose  (Art.  95) 

sin  6  sin  c 

becomes  cos  A  =  ^       .  (2) 

tanc 


Again,  (Art.  97) 

sin  a       sin  c 

in  the  right  triangle  is 

smA  =  *^.  (3) 

sine 

Dividing  (3)  by  (2), 

,,       sin  a  cos  b     sin  a  cos  a  cos  £>          sin  a 
tan  ./i  = 


-  :  —  -      -  :  —  -  -  :  —  -, 

cos  c  sin  b     cos  c  cos  a  sin  6  cos  a  sin  b 

since  cos  a  cos  b  =  cos  0. 

/.  tan  A  sin  6  =  tan  a.  (4) 


124  SPHERICAL   TRIGONOMETRY. 

From  (4)  tan  a  =  tan  A  sin  b, 

also,  tan  b  =  tan  B  sin  a. 

Multiplying,  tan  a  tan  b  =  tan  A  tan  B  sin  a  sin  5, 
or,  cot  A  cot  J5  =  cos  a  cos  &  =  cos  c.  (5) 

From  (2)  and  (3),  by  division, 

tan  b 

cos  A      tan'  £      cos  c 

— : — —  =  — — r  = =  cos  a. 

sin  J5      sin  6      cos  b 

sin  c? 
.-.  cos  A  =  cos  a  sin  B.  (6) 

Let  the  student  write  formulae  (2),  (3),  (4),  (6)  for  B. 
It  will  be  noticed  that  (1)  and  (5)  give  values  for  c  only, 
while  (2),  (3),  (4),  (6)  apply  only  to  A  and  B. 

103.  Formulae  (l)-(6)  are  sufficient  for  the  solution  of 
right  spherical  triangles  if  any  two  parts  besides  the  right 
angle  are  given.  They  are  easily  remembered  by  comparison 
with  corresponding  formulae  in  plane  trigonometry.  Two 
rules,  invented  by  Napier,  and  called  Napier's  Rules  of  Cir- 
cular Parts,  include  all  the  formulae  of  Art.  102. 

Omitting  C,  and  taking  the  comple- 
ments of  A,  c,  and  B,  the  parts  of  the 
triangle  taken  in  order  are  a,  b,  90°  —  A, 
90°  -  c,  90°  -  B.  These  are  called 
the  circular  parts  of  the  triangle. 

Any  one  of  the  five  parts  may  be 
FIG.  SL  selected   as   the   middle  part,  the  two 

parts  next  to  it  are  called  the  adjacent 

parts,  and  the  remaining  two  the  opposite  parts.  Thus,  if 
a  be  taken  as  the  middle  part,  90°  —  B  and  b  are  the 
adjacent  parts,  and  90°  —  c,  90  —  ^4  the  opposite  parts. 


NAPIER'S   RULES.  125 

Napier's  Two  Rules  are  as  follows  : 

The  sine  of  the  middle  part  equals  the  product  of  the  tangents 
of  the  adjacent  parts. 

The  sine  of  the  middle  part  equals  the  product  of  the  cosines 
of  the  opposite  parts. 

It  will  aid  the  memory  somewhat  to  notice  that  i  occurs  in 
sine  and  middle,  a  in  tangent  and  adjacent,  and  o  in  cosine 
and  opposite,  these  words  being  associated  in  the  rules. 

The  value  of  the  above  rules  is  frequently  questioned, 
most  computers  preferring  to  associate  the  formulae  with 
the  corresponding  formulae  of  plane  trigonometry. 

These  rules  may  be  proved  by  taking  each  of  the  parts  as 
the  middle  part,  and  showing  that  the  formulae  derived  from 
the  rules  reduce  to  one  of  the  six  formulae  of  Art.  102. 

Then,  if  b  is  the  middle  part,  by  the  rules, 

sin  b  =  tan  a  tan  (90°  —A)=  tan  a  cot  A,  or  tan  A  = 


sin  o 
sin  b  =  cos  (90°  -  c)  cos  (90°  -  B)  =  sine  sin  B, 


sin  c 


results  which  agree  with  (4)  and  (3),  Art.  102.  If  any 
other  part  be  taken  as  the  middle  part,  the  rules  will  be 
found  to  hold. 

104.   Area  of  the  spherical  triangle. 

If    r  =  radius  of  the  sphere, 

E  =  spherical  excess  of  the  triangle  =A  +  B+  (7  —  180°, 
A  =  area  of  spherical  triangle,  then  by  geometry 


If  the  three  angles  are  not  known,  E  may  be  computed  by 
one  of  the  following  methods,  and  A  found  as  above. 


126  SPHERICAL   TRIGONOMETRY. 

Cagnoli's  Method. 

B  +  C-  180°) 


ri  n 

=  sin  J(J.  +  2?)sin-    —  cos  J(J.  +  -#)cos- 


.  c     c 

sin-  cos- 

=  [cos  £(«-&)-  cos  J  (a  +  *)]  -  -  -     (Art-  101) 

cos- 

„    .    a   .    b  2 

2  sin  -  sin  -         _ 
_  ^       ^     vain  s  sin  (s  —  a)  sin  (s  —  6)  sin  (s  —  c) 

£  sin  a  sin  6 

2  (Arts.  51,  98) 


gjn  ^  _  Vsin  8  sin  (s  -  a)  sin  (8  -  ft)  sin  (8  -  c) 
2  2  cos  |  cos  |  cos  | 

Lhuilier's  Method. 

^=sin^(^  +  ^+  (7-180°) 
4      cos  £(^ +  £+<?- 180°)' 

Now,  multiply  each  term  of  the  fraction  by 
2cosi(4  +  -B-  (7+180°), 
and  by  Art.  51,  (1)  and  (3),  the  equation  becomes 


E 
4 


cos  ^(a  —  b)  —  cos|  c°s^ 

-r in— e  (Art  101) 

cos  J  (a  +  6)  +  cos  -    sm- 

_  sin  l  (g  —  &)  sin  ^  (s  —  a)        I      sin  g  sin  (s  —  c) 
cos  ^  cos  ^  (s  —  c) 

(Art.  51) 


AREA  OF   SPHERICAL   TRIANGLES.  127 

By  Art.  52,  introducing  the  coefficient  under  the  radical, 


I  tan  I  (s  -  a)  tan  |  («  -  6)  tan  |  O  -  c) . 


If  two  sides  and  the  included  angle  are  given,  E  may  be 
determined  as  follows  : 

cos-  =  cos  %(A  +  B  +  C  -  180°) 


=  cos          +       sn+  sn 


cos  \(a  +  ft)  sin2-^  +  cos  J  (a  -  b)  cos2-^    (Art.  101) 


a       b   .     •    a  .    b        n 
cos  -  cos  -  +  sin  -  sin  -  cos  C 

cos 


.    a   .    b    0   .    (7       (7 
sin  -  sin  -  •  2  sin—  cos— 

But      sin^=-  — -•  (Cagnoli's  Method) 

cos| 

Dividing  this  equation  by  the  above, 

sin  -  sin  -  sin  C 
E                    22 
tan—  = 7 T 

cos  -  cos  -  +  sm  -  sm  -  cos  0 

2t  L,  A    >  M 

This  formula  is  not  suitable  for  logarithmic  computations. 
Usually  it  is  better  to  compute  the  angles  by  Napier's  Analo- 
gies, and  solve  by  A  =  Er2  x  — -• 

180 


128  SPHERICAL  TRIGONOMETRY. 


EXAMPLES. 

1.  Show    that    cos  a  =  cos  b  cos  c  +  sin  I  sin  c  cos  A  becomes 

sec  A  =  1  -f  sec  a,  when  a  =  b  =  c. 

2.  If  a  H-  6  +  c  =  IT,  prove 

Tt  S*1 

(a)  cos  a  =  tan  —  tan  — . 


2       sin  b  sin  c 

A 

(c)    sin2  —  =  cot  b  cot  c. 

(</)  cos  A  +  cos  5  -f  cos  C  =  1. 

(e)   sin2  ^  +  sin2 1  +  sin2  ^  =  1. 

sin  —  cos  \  (A  —  E)      sin—  sin  \  (s  —  a) 

3.  Prove -j = —  (Art.  104) 

sin  —  cos  " 

4.  Show  that  cos  a  sin  b  —  sin  a  cos  b  cos  C  +  sin  c  cos  .4. 


CHAPTER   XI. 

SOLUTION    OF    SPHERICAL    TRIANGLES. 

105.  According  to  the  principles  of  spherical  geometry 
any  three  parts  are  sufficient  to  determine  a  spherical  tri- 
angle ;  the  other  parts  are  computed,  if  any  three  are  given, 
by  the  formulae  of  trigonometry.  The  known  parts  may  be  : 

I.    Three  sides,  or  three  angles. 

II.    Two  sides  and  the  included  angle,  or  two  angles  and 
the  included  side. 

III.  Two  sides  and  an  angle  opposite  one,  or  two  angles, 
and  a  side  opposite  one. 

It  will  appear  that,  as  in  plane  geometry,  III  may  be 
ambiguous. 

The  signs  of  the  functions  in  the  formulse  are  important 
since  the  cosines  and  tangents  of  arcs  and  angles  greater 
than  90°  are  negative ;  whether  the  part  sought  is  greater 
or  less  than  90°  is  therefore  determined  by  the  sign  of  the 
function  in  terms  of  which  it  is  found  unless  this  function 
be  sine.  In  this  case  the  result  is  ambiguous,  since  sin  a 
and  sin  (180°  —  a)  have  the  same  sign  and  value.  Thus  if 
the  solution  gives  log  sin  a  =  9.56504,  we  may  have  either 
a  =  21°  33',  or  158°  27'.  The  conditions  of  the  problem 
must  determine  which  values  apply  to  the  triangle  in 
question. 

The  negative  signs,  when  they  occur,  will  be  indicated 

log  cos  115°  20'  -9.63135", 

indicating,  not  that  the  logarithm  is  negative,  but  that 
in  the  final  result  account  must  be  made  of  the  fact  that 
cos  115°  20'  is  negative. 

129 


130  SPHERICAL   TRIGONOMETRY. 

106.   Formulae  for  the  solution  of  triangles. 

j  sin  A  _sin^_sin  C 

sin  a      sin  b      sin  c  * 


II  to    ^i  =    /sin  (8  -  6)  sin  Q  -  c) 

2      ^      smssin(s-a) 


III.  tan|=\/- 


sini(^-  B) 
IV.        ten  i  (a  -  ft)  =  -  f  —  -  tan  f  . 

in  2 


-. 
V.        ten  J  (a  +6)  =  -4-          -tanf. 

cos|(^  +  ^)       2 

sinl(a-6)       c 
VI.      te»U-B)=  —  -       -cot. 


cos  \  (a  -  6)       r 
VII.      tan  |  (^  +  JS)  =  —  J-       -  cot     . 


VIII.  A 

where  ^  is  determined  by 


Right  triangles  may  be  solved  as  special  cases  of  oblique 
triangles,  or  by  the  following  : 

(1)  cos  c  =  cos  a  cos  6  .  (4)    tan  A  sin  b  =  tan  a. 

(2)  cos  .1={^.  (5)    cot  A  cot  B  =  cose. 


(3)    sin^--  (6)    cos  A  =  cos  a  sin  B. 

S1H  C 

The  formula  to  be  used  in  any  case  may  be  determined  by 
applying  Napier's  Rule  of  Circular  Parts. 

107.  In  solving  a  triangle  the  student  should  select  formulae 


MODEL   SOLUTIONS.  131 

in  which  all  parts  save  one  are  known,  and  solve  for  that 
one  (see  page  77).  Referring  to  Arts.  105  and  106,  it  will 
appear  that  solutions  are  effected  as  follows  : 

Case  I  by  formulae  II,  or  III,  check  by  I. 
Case  II  by  formula  VI,  VII,  I,  or  IV,  V,  I,  check  by  IV 
or  VI. 

Case  III  by  formulse  I,  IV,  or  I,  VI,  check  by  VI  or  IV. 

MODEL   SOLUTIONS. 
108.  1.   Given  a  =  46°  24',  b  =  67°  14',  c  =  81°  12'.     Solve. 


tan  -  =  Jsin  0  ~  ft)  sin  0  ~  c\   tan  -  =  Jsin  (•*  -  a)  sin  0*  - 
2      ^       sin  s  sin  (s  —  a}  2       *       sin  s  sin  (s  —  b) 


tan  -  =  Jsin  C*  ~  q)  sin  (*  -  ft)  .       Check  :   sin  a  =  sin  b  - 
2       *        sin  s  sin  (s  —  c)  sin  A      sin  B 

Arrange  and  solve  as  in  Example  1,  page  80. 

Aru.  A  =  46°  18'.5,  B  =  ,    C  = 

Solve  :         (1)    A  =  96°  45',     B  =  108°  30',    C  =  116°  15'. 
(Use  formulse  III  in  the  same  manner  as  in  Example  1.) 

(2)  a  =  108°  14',    b  =  75°  29',       c  =  56°  37'. 

(3)  A  =  57°  50',     B  =  98°  20',      C  =  63°  40'. 

2.    Given  b  =  113°  3',  c  =  82°  39',   A  =  138°  50'.     Solve. 


2  sin  |(^  -  C) 

b  =  113°   3'     log  cos  i  (b  -  c)  =  9.98453        log  sin  \(b-  c)  =  9.41861 
c=  82°  39'  colog  cos  i  (b  +  c)  =0.86461"  colog  sin  £  (b  +  c}  =0.00409 

i(6  +  c)=  97°  51'  log  cotl  =9.57466  log  cot  -  =  9.57466 

l(b-c)=   15°  12'  2  2     _ 

i  ^  _  69o  25'  log  tan  |  (B  +  C)  =  0.42380"  log  tan  £  (J3  -  C)  =  8.99736 
^(jB-f  C)=110°  39'  i(5-C)  =  5°40/.6 

j(jB-C)=     5°40'.6 
.-.  JS  =  116°19'.6 
and  C  =  104°58'.4 


132  SPHERICAL   TRIGONOMETRY. 

Check: 

log  sin  A  =  9.81839  log  tan  %  (b  -  c)  =  9.43408 

log  sin  b  =  9.96387  log  sin  $  (£  +  C)  =  9.97116 

cologs  in  5  =  0.04756  colog  sin  1(3-  C)  =  1.00474 

log  sin  a  =  9.82982  log  tan  «  _  Q.40998 

a  =  137°  29' 

a  =  137°  29' 

Notice  that  tan  h  (B  +  (7)  is  — .    Hence,  £  (5  +  (7) is  greater  than  90°,  i.e.  110°  3y. 

Solve:         (1)    A  =    68° 40',   £  =    56° 20',     c=    84° 30'. 
(Use  formulae  IV,  V,  I.    Compare  Example  2.) 

(2)  a  =  102°  22',    b  =    78°  17',    C  =  125°  28'. 

(3)  A  =  130°    5',   5  =    32°  26',     c  =    51°    6'. 

109.  Ambiguous  cases.  By  the  principles  of  geometry  the 
spherical  triangle  is  not  necessarily  determined  by  two  sides 
and  an  angle  opposite,  nor  by  two  angles  and  a  side  opposite. 
The  triangle  may  be  ambiguous.  By  geometrical  principles 
it  is  shown  that  the  marks  of  the  ambiguous  spherical  tri- 
angle are: 

1.  The  parts  given  are  two  angles  and  the  side  opposite 
one,  or  two  sides  and  the  angle  opposite  one. 

2.  The  side,  or  angle,  opposite  differs  from  90°  more  than 
the  other  given  side,  or  angle. 

3.  Both  sides,  or  angles,   given  are  either  greater  than 

90°,  or  less  than  90°. 

In  the  right  triangle  AB  (72, 

sin  a  =  sin  A  sin  c.   (formula  (3)) 

Therefore  there  will  be  no  solution,  one 
solution,  or  two  solutions,  according  as 
sin  a  =  sin  A  sin  <?,  i.e.  according  as  a  = 
the  perpendicular  JP.  (See  Art.  65.) 
But  the  most  expeditious  means  of  determining  the  am- 
biguity is  found  in  the  solution  of  the  triangle.  The  use 
of  formula  I  gives  the  solution  in  terms  of  sine,  so  that  it  is 
to  be  expected  that  two  values  of  the  part  sought  may  be 
possible ;  and  whether  the  triangle  be  ambiguous  or  not, 
there  must  be  some  means  of  determining  which  of  the  two 


AMBIGUOUS   SPHERICAL   TRIANGLES.  133 

angles,  a  and  180°  —  a,  that  have  the  same  sine  is  to  be  used. 
If  there  are  two  solutions,  both  values  are  used. 

This  is  determined  in  the  further  solution  of  the  triangle 
by  formula  VI,  which  may  be  written 

b  =  cos  j-  (A  +  O)  tan  -|-  (a  +  c) 
'    2  cos  J  (A  -  0) 

Now  —  <  90°,  whence  tan  —  is  +  .     Then  if  for  both  values 

LJ  2i 

of  (7,  found  by  the  sine  formula,  the  second  member  is  -f  , 
there  are  two  solutions  ;  if  the  second  member  is  —  for 
either  value  of  (7,  there  is  but  one  solution  ;  while  if  both 
values  of  (7  make  the  second  member  —  ,  there  is  no  solution. 
The  various  cases  will  be  illustrated  by  problems. 

3.    Given  a  =  62°  15'.4,   b  =  103°  18'.8,   A  =  53°  42'.6.     Solve. 


tan     = 


B)  tan  \  (a 


sin  a  2  cos  ^  (A  —  B) 

sin  C  =  sincsin^       Check  :  cot  ^  =  tan        *  -&    *™ 


sin  a  2  sin  \  (a  —  b) 

Solving  the  first  formula  gives 

log  sin  £  =  9.94756, 

whence  Bl  =  62°  24'.4, 

B2  =  117°  35'.6. 
For  each  of  the  values  Bl  and  B.2, 

cos  |  (A  +  £)  tan  1  (q  +  6) 
cos  \  (A  -  B) 

is  +  and  therefore  equal  to  tan  |-     Hence  there  are  two  solutions.     Find 

c  =  153°   9'.6,  or  70°  25'.4 
and  C  =  155°  43'.2,  or  59°    6'.2 

4.   Given  a  =  46°45'.o,   A  =  73°  ll'.S,   5  =  61°  18'.2.     Solve. 

cot  ^  =  tan  K^  -  #)  cos  ^  (a  +  6) 


sin  ^4  2  cos  i  (a  -  6) 


sin  c  =  .        Check  :  tan  £  =  tan  j  («  -  *)  sin}  (^  + 

sin  ^4  2  sin  £  (/I  -  5) 


134  SPHERICAL   TRIGONOMETRY. 

Solving  for  b  gives  log  sin  b  =  9.82446, 

whence  ^  =    41°  52'.5, 

and  62  =  138°    7'.5. 

For  the  value  6:  the  fraction 

tan  I  (A  -  B)  cos  ^  (a  +  6) 
cos  £  (a  —  6) 

is  +,  but  for  62  cos  ^  (a  +  b)  is  — ,  making  the  fraction  — ,  and  hence  it 

C* 
can  not  equal  cot  — ,  which  is  +.     There  is  then  but  one  solution.    Find 

C  =  60°  42'.7,   c  =  41°  35M. 

5.   Given  a  =  162°  30',   A  =  49°  50',   B  =  57°  52'.     Solve. 
Solving  gives  log  sin  b  =  9.52274, 

whence  \=    19°  27'.9, 

b2  =  160°  32M. 
For  both  values,  &j  and  52,  cos  |  (a  -f  b)  is  — .     Therefore, 

tan  l(A  -  B}  cos  j  («  +  6) 
cos  ^  (a  —  6) 

ri 

is  —  and  not  equal  to  cot  — •     Hence  the  triangle  is  impossible. 

Solve,  testing  for  the  number  of  solutions  : 

(1)  &  =  106°24'.5,     c=    40°  20',  C=    38°  45'.6. 

(2)  a=    80°  50,       4=131°  40',  B  =    65°  25'. 

(3)  a  =    60°  31 '.4,     b  =  147°  32'.  1,  B  =  143°  50'. 

(4)  a  =    55°  30',        c  =  139°    5',  A  =    43°  25'. 

RIGHT   TRIANGLES. 

110.  Right  triangles  are  a  special  case  of  oblique  triangles, 
but  are  usually  solved  by  formulae  (1)  to  (6),  Art.  106. 
Students  should  have  no  difficulty  in  applying  these. 

Computers  generally  question  the  utility  of  Napier's  Rules- 
of  Circular  Parts.  For  those  who  prefer  the  rules  a  problem 
will  be  solved  by  their  use. 


SPECIES.  135 

6.    Given  c  =  86°  51',   B  =  18°  3'.5,     C  =  90°. 

The  parts  sought  are  a,  6,  A,  and  it  is  immaterial  which  is  computed 
first,  a  and  A  are  adjacent  to  c  and  B,  while  b  is  the  middle  part  of  c 
and  jB.  Then  by  Napier's  first  rule 

f\/\O       T> 

sin  (90°  -  £)  =  tan  (90°  -  c)  tan  a  ; 

or  tan  a  =  c-^^  =  cos  B  tan  c, 

cote 

which  is  formula  (2). 
By  the  same  rule 

sin  (90°  -  c)  =  tan  (90°  -A)  tan  (90°  -  B), 


or  cot  A  =       -2-  -  cos  c  tan  5,  formula  (5). 

cot  B 

Finally  by  the  second  rule 

sin  b  =  cos  (90°  -  c)  cos  (90°  -  B)  =  sin  c  sin  B,      formula  (3). 
The  solutions  give  a  =  86°  41  '.2,   b  =  18°  1'.8,   A  =  88°  58'.4.    Verify. 

111.  Species.  Two  angles  or  sides  of  a  spherical  triangle 
are  said  to  be  of  the  same  species  if  they  are  both  less,  or 
both  greater,  than  90°.  They  are  of  opposite  species  when  one 
is  greater  and  the  other  less  than  90°.  Since  the  sides  and 
angles  of  a  spherical  triangle  may,  any  or  all,  be  less  or 
greater  than  90°,  it  is  necessary  in  solutions  to  determine 
whether  each  part  is  more  or  less  than  909.  The  directions- 
already  given  are  sufficient  in  oblique  triangles.  In  right 
triangles  the  sign  of  the  function  will  determine  if  the  solu- 
tion gives  the  result  in  terms  of  cosine  or  tangent,  but  not 
if  the  result  is  found  in  terms  of  sine.  Thus  in  Example  6, 
above,  we  have  log  sin  b  =  9.49068,  whence  b  =  18°  I/.  8,  or 


161°  58'.  2.     By  formula  (4)  sin  6  =  Now  sin  b  is 

tan  A 

always  +  ,  therefore,  tan  a  and  tan  A  must  be  of  the  same 
sign,  whence  in  any  right  spherical  triangle  an  oblique  angle 
and  its  opposite  side  must  be  of  the  same  species. 

Again  by  formula  (1)  cos  c  =  cos  a  cos  b.  Now  cos  c  is  -f 
or  —  according  as  c  is  less  or  greater  than  90°.  If  then 
<?<90°,  cos  a  and  cos  b  are  of  the  same  sign,  but  if  <?>90°, 
cos  a  and  cos  b  are  of  opposite  sign.  Therefore,  if  the 


136  SPHERICAL   TRIGONOMETRY. 

hypotenuse  of  a  right  spherical  triangle  is  less  than  90°,  the 
other  sides,  and  hence  the  angles  opposite,  are  of  the  same 
species ;  but  if  the  hypotenuse  be  greater  than  90°,  the  other 
sides,  and  the  angles  opposite,  are  of  opposite  species. 

112.  Ambiguous  right  triangles. 

When  the  parts  given  are  a  side  adjacent  to  the  right 
angle,    and   the    angle    opposite   this   side,   the  triangle   is 
ambiguous,   for    solving   for   the   hypot- 
enuse by  formula  (3)  gives 


OAii     V, —  — , 

sin  A 

FIG.  54.  from  which  there  result  two  values  of  c. 

By  the  last  rule  of  species  it  follows  that 
to  the  values  of  c,  one  <90°,  the  other  >90°,  there  will  cor- 
respond two  values  for  b,  one  of  the  same  species  as  a,  the 
other  of  opposite  species. 

Clearly  sin  c  §  1,  according  as  sin  a  =  sin  A,  and  hence 
there  will  be  no  solution,  one  solution,  or  two  solutions, 
according  as  sin  a  =  sin  A. 

Solve  the  spherical  triangles,  right  angled  at  C,  given : 

(1)  b  =  73°  21'. 4,    c=  84°  48'. 7. 

(2)  c  =  54°  28',   B  =  128°  12'.6. 
...  (3)  b  =  45°  42',   B  =  135°  42'. 

(4)  a  =108°  22'. 3,    b  =  120°  14'. 5. 

(5)  a  =  70°  50',   A  =  170°  40'. 

(6)  b  =  32°  8'. 4,   .£=46°  2'. 8. 

(7)  b  =  34°  28',   c  =  62°  50'. 

(8)  c  =  102°  35',   B  =  17°  45'. 

(9)  a  =92°  16',    <?=57°  35'. 


FIVE-PLACE 
LOGARITHMIC  AND  TRIGONOMETRIC 

TABLES 


ADAPTED  FROM  GAUSS'S  TABLES 

BY 

ELMER   A.    LYMAN 

MICHIGAN   STATE   NORMAL   COLLEGE 
AND 

EDWIN   G.    GODDARD 


UNIVERSITY    OF   MICHIGAN 


ALLYN    AND    BACON 
ant) 


COPYRIGHT,  1899,  BY 
ELMER  A.  LYMAN  AND 
EDWIN  C.  GODDARD. 


J.  8.  Gushing  &  Co.  —  Berwick  &  Smitti 
Norwood  Mase.  U.S.A. 


TABLE   I. 

THE  COMMON  LOGARITHMS  OF  NUMBERS 
FROM  1   TO  10009. 


N. 

L.  0       1       2       3       4 

56789 

P.P. 

100 

101 
102 
103 
104 

oo  ooo    043    087     130    173 
432    475    5i8    561    604 
860    903    945;    988  #030 
01  284    326    368    410    452 
703    745    787     828     870 

217    260    303    346    389 
647    689    732    775    817 
#072  #115  #157  *i99  3,242 
494    53^    578     620    662 
912    953    995  ^036  ^078 

i 

2 

3 

4 

7 
8 

9 

i 

2 

3 

4 

7 
8 

9 

i 

9 

i 

2 

3 
4 
5 
6 

7 
9 

i 

2 

3 

4 

I 

44     43     42 

4,4     4,3     4,2 
8,8     8,6     8,4 
13,2    12,9   I2,6 
17,6   17,2   16,8 

22,0    21,5     2I,° 

26,4  25,8  25,2 
30,8  30,1  29,4 

35,2  34,4  33,6 
39,6  38,7  37,8 

41     40     39 

4,1     4,o     3,9 
8,2     8,0     7,8 
12,3   I2,°   IIi7 
16,4   16,0   15,6 
20,5   20,0   19,5 
24,6  24,0  23,4 
28,7   28,0   27,3 
32,8   32,0  31,2 
36,9  36,0  35,1 

38     37     36 

3,8     3,7     3,6 
7,6     7,4     7,2 
11,4   II,1    I0,8 
15,2   14,8    14,4 
19,0  18,5   18,0 

22,8     22,2    21,6 

26,6   25,9   25,2 
30,4   29,6   28,8 
34,2  33,3   32,4 

35     34     33 

3,5     3,4     3,3 
7,0     6,8     6,6 
10,5    I0,2     9,9 
14,0   13,6   13,2 
17,5   17,0  16,5 
21,0  20,4   19,8 
24,5   23,8   23,1 
28,0  27,2  26,4 
31,5  30,6  29,7 

32     31     30 

3,2     3,i     3,o 
6,4     6,2     6,0 
9,6     9,3     9,o 
12,8    12,4   I2,o 
16,0   15,5   15,0 
19,2   18,6   18,0 
22,4  21,7   21,0 
25,6   24,8   24,0 
28,8   27,9   27,0 

I°5 

106 
107 
108 
109 

02  119      160      202      243      284 

531     572    612    653     694 
938    979  *oi9  *o6o  *ioo 
03342    383    423    463     503 
743    782     822     862    902 

325     366    407    449    490 
735    776     816     857     898 

#I4I    #l8l    *222   ^262   *302 

543     583     623    663    703 

941      981    *O2I    #060   *IOO 

110 

in 

112 

"3 
114 

04  139     179    218     258     297 
532    571     610    650    689 
922    961     999  ^038  *077 
05308    346    385    423    461 
690    729    767     805     843 

336    376    415    454    493 
727    766    805     844    883 
*ii5  *i54  ^192  3,231  ^269 
500    538    576    614    652 
881     918    956    994  ^032 

115 

116 
117 
118 
119 

06  070      I08       145       183      221 

446    483    521    558    595 
819,  856    893    930    967 
07  188    225    262    298    335 
555    59i    628    664    700 

258     296    333     371     408 
633     670    707    744    781 
*004  *04i-  ^078  *H5  #151 
372    408    445     482    518 
737    773     809     846     882 

120 

121 
122 
123 
124 

918    954    990  *027  ^063 
08  279    314    350    386    422 
636    672    707    743    778 
991  ^026  *o6i  $096  ^132 
09342    377    412    447    482 

*099  *I35  *i7i  *207  *243 
458    493    529    565    600 
814    849    884    920    955 
^167  *202  *237  ^272  ^307 
517     SS2    587     621     656 

125 
126 
I27 
128 
129 

691     726    760    795     830 
10  037    072     106     140     175 
380    415    449    483     517 
721    755    789    823    857 
ii  059    093     126    160    193 

864    899    934    968  #003 
209     243     278     312    346 
551     585     619     653    687 
890    924    958    992  ^025 
227    261    294    327    361 

130 

131 
132 

133 

134 

394    428    461     494    528 
727    760    793     826     860 
12  057     090     123     156     189 
385    418    450    483     516 
710    743    775     808     840 

561    594    628    66  i    694 
893    926    959    992  *Q24 

222      254      287      32O      352 
548      581      613      646      678 

872    905    937    969  *ooi 

135 

137 
138 
139 

13  033    066    098     130     162 
354    386    418    450    481 
672    704    735    767    799 
988  ^019  $05  i  ^082  ^114 
14301    333    364    395    426 

194    226    258    290    322 
5i3    545    577    609    640 
830    862    893    925    956 
*I45  ^176  *2o8  *239  *27o 
457    489    520    551    582 

140 

141 
142 

143 

144 

613    644    675    706    737 
922    953    983  *oi4  ^045. 
15  229    259    290    320    351 
534    564    594    625    655 
836    866    897    927    957 

768    799    829    860    891 
^076  *io6  *I37  *i68  ^198 
381     412    442    473     503 
685     7*3     746     776     8o6 
987  *oi7  *047  *077  *io7 

145 

146 

147 
148 
149 

16  137     167     197    227    256 

435    465     495     524    554 
732    761     791     820    850 
17  026    056    085     114     143 
319    348    377    406    435 

286     316    346    376    406 
584    613     643     673    702 
879    909    938     967    997 
173     202    231     260     289 
464    493     522    551     580 

150 

609    638    667    696    725 

754    782    811    840    869 

N. 

L.  0       1       2       3       4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

150 

151 
152 
153 
154 

17  609  638  667  696  725 
898  926  955  984  #013 
18  184  213  241  270  298 
469  498  526  554  583 
752  780  808  837  865 

754  782  811  840  869 
*04i  *070  *099  *I27  #156 
327  355  384  412  441 
611  639  667  696  724 
893  921  949  977  #005 

i 

2 

3 

29   28 

2,9   2,8 

*',7   8,'4 

g 

% 

159 

19  033  061  089  117  145; 
312  340  368  396  424 
590  618  64^  673  700 
866  893  921  948  976 
20  140  167  194  222  249 

173   201   229   257   285 

45i  479  5°7  533  562 
728  756  783  811  838 
*oo3  ^030  ^058  *o8£  #112 
276  303  330  358  385 

4 

7 
8 

9 

11,6   11,2 

14,5  14,0 

17,4   16,8 
20,3   J9,6 
23,2   22,4 
26,1   25,2 

160 

161 
162 
163 
164 

412  439  466  493  520 
683  710  737  763  790 
952  978  *oo$  ^032  *059 
21  219  245  272  299  325 
484  511  537  564  590 

548  575  602  629  656 
817  844  871  898  925 
*o8S  #ii2  *i39  *i6s  #192 
352  378  405  431  458 
617  643  669  696  722 

I 

2 

3 

27   26 

2,7   2,6 
5,4   5/2 
8,1   7,8 

165 
166 
167 
168 
169 

748  775  801  827  854 

22  01  1   037   063   089   115 
272   298   324   350   376 

53i  557  583  608  634 
789  814  840  866  891 

880  906  932  958  985 

141   167   194   220   246 

401  427  453  479  505 
660  686  712  737  763 
9J7  943  968  994  *OI9 

4 
9 

10,8   10,4 

13,5   13,0 
16,2   15,6 
i8;9   18,2 

21,6   20,8 

24,3   23,4 

170 

171 
172 
173 
174 

23  045  070  096  121  147 
300  325  350  376  401 
553  578  603  629  654 
805  830  855  880  905 
24055  080  105  130  155 

172  198  223  249  274 
426  452  477  502  528 
679  704  729  754  779 
930  953  980  #005  ^030 
i  80  204  229  254  279 

25 

i  2,5 
2  5,0 
3  7,5 

i75 
176 
177 
178 
179 

304  329  353  378  403 
551  576  601  62^  6^0 
797  822  846  871  895 
25  042  066  091  115  139 
285  3io  334  358  382 

428  452  477  502  527 
674  699  724  748  773 
920  944  969  993  #018 
164  188  212  237  261 
406  431  455  479  503 

4  IO/° 
5  12,5 
6  15,0 

7  17,5 
8  20,0 
9  1  22,5 

180 

181 
182 
183 
184 

527  55i  575  600  624 
768  792  816  840  864 
26007  031  055  079  102 
24^  269  293  316  340 
482  505  529  553  576 

648  672  696  720  744 
888  912  935  959  983 

126   150   174   198   221 

364  387  .  411  435  458 
600  623  647  670  694 

i 

2 

3 

24   23 

2,4   2,3 
4,8   4,6 
7,2   6,9 

185 
186 
187 
188 
189 

717  741  764  788  811 
95i  973  998  *Q2i  *o45 
27  184  207  231  254  277 
416  439  462  485  508 
646  669  692  715  738 

834  858  88  I  905  928 
#068  #09  1  *H4  #138  #161 
300  323  346  370  393 
53i  554  577  600  623 
761  784  807  830  852 

4 

I 
I 

9 

9,6   9,2 
12,0   11,5 
14.4   13,8 
16,8   16,1 
19,2   18,4 
21,6   20,7 

190 

191 
192 

193 
194 

875  898  921  944  967 
28  103  126  149  171  194 
330  353  37$  398  421 
556  578  601  623  646 
780  803  825  847  870 

989  *oi2  3,035  #058  *o8i 
217  240  262  285  307 
443  466  488  511  533 
668  691  713  735  758 
892  914  937  959  981 

i 

2 
3 

22   21 

2,2    2,1 
4,4    4,2 

6,6   6,3 

195 
196 
197 
198 
199 

29  003  026  048  070  092 
226  248  270  292  314 
447  469  491  513  535 
667  688  710  732  754 
88^  907  929  951  973 

115  137  159  181  203 
336  358  380  403  425 
557  579  601  623  645 
776  798  820  842  863 
994  *oi6  ^038  *o6o  *o8  i 

4 

1 

7 
8 

9 

8,8   8,4 
11,0   10,5 
13,2   12,6 

15,4   14,7 
17,6   16,8 
19,8   18,9 

200 

30  103  125  146  168  190 

211  233  255  276  298 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

200 

201 
202 
203 
204 

30  103  125  146  168  190 

320  341  363  384  406 
535  557  578  600  621 
750  771  792  814  835 
963  984  *oo6  #027  ^048 

211   233   255   276   298 

428  449  471  492  514 
643  664  68£  707  728 
856  878  899  920  942 
#069  #091  *ii2  #133  #154 

M 

I   2 

2   4 

3  6 
4  8 
5  « 
6  13 
7  15 
8  17 

9  19 

I 

2 

3 
4 
5 
6 

1 

9 

i 

2 

3 
4 

7 
8 

9 

I 

2 

3 
4 

7 
8 

9 

2 

3 
4 

7 
8 

9 

\   21 

2     2,1 

,4    4/2 

,6   6,3 
8   8,4 
0   IO/5 

2    12,6 

4   14/7 
6   16,8 
8   18,9 

20 

2,0 
4/o 
6,0 
8,0 

10,0 
12,0 
14,0 

16,0 
18,0 

19 

ST. 

% 

9/5 
"/4 
13/3 

J5/2 

17/1 

18 

1/8 
3,6 
5/4 
7/2 
9/o 
10,8 

12,6 

14/4 
16,2 

17 

i/7 
3/4 
5/i 
6,8 

8/5 

IO,2 

n,9 

13/6 

J5/3 

207 
208 
209 

31  175  197  218  239  260 
387  408  429  450  471 
597  618  639  660  681 
806  827  848  869  890 
32015  035  056  077  098 

281  302  323  345  366 

492  513  534  555  576 
702  723  744  765  785 
911  931  952  973  994 
118  139  160  181  201 

210 

211 
212 
213 
2I4 

222   243   263   284   305 

428  449  469  490  510 
634  654  675  695  715 
838  858  879  899  919 

33  041   062   082   102   122 

325  346  366  387  408 
53i  552  572  593  613 
736  756  777  797  818 
940  960  980  *ooi  #021 
143  163  183  203  224 

215 

216 

217 

218 

219 

244   264   284   304   325 

445  465  486  506  526 
646  666  686  706  726 
846  866  885  905  925 
34  044  064  084  104  124 

345  365  385  405  425 
546  566  586  606  626 
746  766  786  806  826 
945  965  985  *oo5  *025 
143  163  183  203  223 

220 

221 
222 
223 
224 

242  262  282  301  321 
439  459  479  498  518 
635  655  674  694  713 
830  850  869  889  908 
35  025  044  064  083  102 

341  361  380  400  420 
537  557  577  596  616 
733  753  772  792  811 
928  947  967  986  *oo$ 
122  141  160  180  199 

225 
226 
227 

228 
229 

218  238  257  276  295 
411  430  449  468  488 
603  622  641  660  679 
793  813  832  851  870 
984  #003  #021  #040  *059 

3i5  334  353  372  392 
507  526  545  564  583 
698  717  736  755  774 
889  908  927  946  965 
#078  *097  *n6  #135  #154 

230 

231 
232 
233 

234 

36  173   192   211   229   248 

361  380  399  418  436 
549  568  586  605  624 
736  754  773  79i  8  10 
922  940  959  977  996 

267  286  305  324  342 

455  474  493  5"  53O 
642  661  680  698  717 
829  847  866  884  903 
*oi4  *033  ^051  ^070  *o88 

235 
236 

237 
238 
239 

37  107  125;  144  162  181 
291  310  328  346  365 

475  493  5"  530  548 
658  676  694  712  731 
840  858  876  894  912 

199  218  236  254  273 
383  401  420  438  457 
566  585  603  621  639 
749  767  785  803  822 
93  i  949  967  985  *°°3 

240 

241 
242 

243 
244 

38021  039  057  075  093 

202   220   238   256   274 

382  399  417  435  453 
561  578  596  614  632 
739  757  775  792  810 

112  130  148  166  184 
292  310  328  346  364 
471  489  507  525  543 
650  668  686  703  721 
828  846  863  881  899 

245 
246 
247 
248 
249 

917  934  952  970  987 
39  094  in  129  146  164 
270  287  305  322  340 
445  463  480  498  515 
620  637  655  672  690 

*oo£  *023  #041  ^058  #076 
182  199  217  235  252 
358  375  393  4io  428 
533  55o  568  585  602 
707  724  742  759  777 

250 

794  8n  829  846  863 

881  898  915  933  950 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.   0       1       2       3       4 

56789 

I 

.P. 

250 

251 
252 

253 

254 

39  794     811     829     846    863 
967     985  #002  #019  #037 
40  140     157     175     192     209 
312    329     346     364    381 
483    500    518    535    552 

881    898    915    933    950 
#054  #07  i  *o88  #106  #123 
226     243     261     278     295 
398    415    432    449    466 
569    586    603     620    637 

2 

3 

18 

1/8 
3-6 

5/4 

255 
256 

257 
258 

259 

654    671     688     705     722 
824     841     858     875     892 
993  #010  #027  #044  #061 

41   l62       179       196      212      229 

33°    347     363     38o    397 

739     756    773    790    807 
909    926    943     960    976 
#078  #095  *n  i  #128  #145; 
246    263    280    296    313 
414    430    447    464    481 

4 
5 
6 

I 

9 

7,2 

9/o 
10,8 

12,6 

H/4 

l6,2 

260 

261 
262 
263 
264 

497     5H    53i     547     564 
664    681     697     714    731 
830    847     863     880     896 
996  #012  #029  #045  #062 
42  160     177     193     210    226 

581    597    614    631    647 
747    764    780    797     814 
913     929    946    963    979 
#078  #095  *n  i  #127  #144 
243     259     275     292*308 

i 

2 

3 

17 

i/7 
3/4 

K 

265 
266 
267 
268 
269 

325    34i    357    374    390 
488    504    521    537    553 
651    667    684    700    716 
813    830    846    862    878 
975    991  #008  #024  #040 

406    423    439    455    472 
570    586    602    619    635 

732    749    765    781     797 
894    911     927     943     959 
#056  #072  #088  #104  #120 

4 
5 
6 

7 
8 

9 

6,8 
8/5 

10,2 

«,9 

13/6 
15/3 

270 

271 
272 
273 
274 

43  136     152     169     185    201 
297    313     329    345    36i 
457    473    489    5°5    52i 
616    632    648     664    680 
775    791     807    823    838 

217     233     249     265     281 
377     393    4°9    425    441 
537    553    569    584    600 
696    712    727    743    759 
854     870    886    902    917 

i 

2 

3 

16 

1/6 
S/2 
4,8 

*75 
276 
277 
278 
279 

933     949    965    98i     996 
44  091     107     122     138     154 

248   264   279   295  311 
404   420   436   451    467 
560   576   592    607    623 

#012  #028  #044  #059  *c>75 
170     18^     201     217     232 
326    342    358    373    389 
483    498    514    529    545 
638    654    669    685    700 

4 

7 
8 

9 

6/4 
8,0 
9/6 

11,2 
12,8 

H/4 

280 

281 
282 
283 
284 

716    731     747     762    778 
871     886    902    917     932 
45  025    040    056    071    086 
179     194    209    225    240 
332    347    S62    378    393 

793     809     824    840     855 
948     963    979    994  #010 
102     117     133     148     163 
255     271     286     301     317 
408     423    439    454    469 

i 

2 

3 

15 

i/5 
3/o 
4/5 

285 
286 
287 
288 
289 

484    500    515    530    545 
637    652    667    682    697 
788     803     818     834     849 
939    954    969    984  *ooo 
46090     105     120     135     150 

561     576     591     606    621 
712    728     743    758     773 
864     879     894    909    924 
#015  #030  #045  #060  #075 
165     180     195     210     225 

4 

I 

7 
8 

9 

6/0 
7/5 
9/o 
10,5 

12,0 

13/5 

290 

291 
292 

293 
294 

240     255     270    285    300 
389    404    419    434    449 
538    553    568    583    598 
687    702    716    731    746 
835    850    864    879    894 

3i5    33°    345    359    374 
464    479    494    509    523 
613     627    642     657     672 
761     776    790     80$     820 
909    923    938    953    967 

i 

2 

3 

14 

i/4 

2,8 
4/2 

295 
296 
297 
298 
299 

982    997  #012  #026  #041 
47  129     144     159     173     188 
276     290    305     319    334 
422    436    451     46^     480 
567     582    596    611     62=; 

#056  #070  #085  #100  #114 

202      217      232      246      26l 

349    363    378    392    407 
494    509    524    538    553 
640    654    069    683    698 

4 
5 
6 

? 

9 

5/6 
7/o 
8/4 
9/8 

11,2 
12,6 

300 

712    727     741     756    770 

784    799    813    828    842 

N. 

L.   0       1       2       3       4 

56789 

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P.P. 

N. 

L.  0   1   2   3   4 

56789 

P 

.P. 

300 

301 
302 
303 
3°4 

47712  727  741  756  770 
857  871  885  900  914 
48  ooi  oic;  029  044  058 

144   159   173   187   202 

287  302  316  330  344 

784  799  813  828  842 
929  943  958  972  986 
073  087  101  116  130 
216  230  244  259  273 
359  373  387  401  4i6 

j 

15 

i  c; 

3°5 
306 
307 
308 
309 

430  444  458  473  487 
572  586  601  615  629 
714  728  742  756  770 
855  869  883  897  911 
996  *oio  #024  ^038  ^052 

501  515  530  544  558 
643  657  671  686  700 
783  799  813  827  841 
926  940  954  968  982 
*o66  *o8o  #094  *io8  *I22 

2 

3 

4 

7 

3,o 

& 

7,5 
9,o 
10,5 

310 

311 
312 

3i3 

3H 

49  136  150  164  178  192 
276  290  304  318  332 
415  429  443  457  471 
554  568  582  596  6  10 
693  707  721  734  748 

206   220   234   248   262 

346  360  374  388  402 
485  499  513  527  541 
624  638  651  665  679 
762  776  790  803  817 

8 
9 

12,0 
13,5 

3i5 
316 

3i7 
3i8 
3i9 

831  843  859  872  886 
969  982  996  *oio  ^024 
50  106  120  133  147  161 
243  256  270  284  297 
379  393  406  420  433 

900  914  927  941  953 
*037  *05  i  *o63  *079  ^092 
174  i  88  202  215  229 

311  323  338  352  365 

447  461  474  488  501 

i 

2 

3 
4 

14 

i,4 
2,8 

4,2 
5,6 

320 

321 
322 
323 
324 

515  529  542  556  569 
651  664  678  691  705 
786  799  813  826  840 
920  934  947  961  974 
51  053  068  08  1  093  108 

583  596  610  623  637 
718  732  745  759  772 
853  866  880  893  907 
987  *ooi  #014  #028  ^041 
121  133  148  162  173 

I 

9 

7,o 
8,4 
9,8 

11,2 
12,6 

P 
2 

329 

188  202  215  228  242 
322  335  348  362  375 
453  468  481  493  508 
587  601  614  627  640 
720  733  746  759  772 

255  268  282  29^  308 
388  402  413  428  441 
521  534  548  561  574 
654  667  680  693  706 
786  799  812  825  838 

i 

2 

13 

J,3 

2  6 

330 

33i 

332 
333 
334 

851  863  878  891  904 

983   996  *009  *022  *03$ 

52  114  127  140  153  166 
244  257  270  284  297 
373  388  401  414  427 

917  930  943  957  970 
#048  *o6i  #073  #088  #101 
179  192  205  218  231 
310  323  336  349  362 
440  453  466  479  492 

3 
4 

7 

8 

3,9 

7/3 
9,i 
10,4 

335 
336 
337 
338 
339 

504  517  530  543  556 
634  647  660  673  686 
763  776  789  802  813 
892  903  917  930  943 
53020  033  046  058  071 

569  582  595  608  621 
699  711  724  737  750 
827  840  853  866  879 
956  969  982  994  #007 
084  097  no  122  135 

9 

n,7 
12 

340 

34i 
342 
343 
344 

148  161  173  186  199 
275  288  301  314  326 
403  415  428  441  453 
529  542  553  567  580 
656  668  68  i  694  706 

212   224   237   230   263 

339  352  364  377  390 
466  479  491  504  517 
593  605  618  631  643 
719  732  744  757  769 

i 

2 

3 
4 

5 
g 

1,2 
2,4 

3,6 
4,8 
6,0 

345 
346 
347 
348 
349 

782  794  807  820  832 
908  920  933  945  958 
54033  045  058  070  083 
158  170  183  195  208 
283  293  307  320  332 

843  857  870  882  895 
970  983  995  *oo8  *020 
095  108  120  133  145 
220  233  245  258  270 
345  357  370  382  394 

9 

8,4 
9,6 
10,8 

350 

407  419  432  444  456 

469  481  494  506  518 

N. 

L.  0   1   2   3   4 

56789 

I 

>.  P. 

N. 

L.   0       1       2       3       4 

56789 

I 

.P. 

350 

35i 
352 
353 
354 

54407    419    432    444    456 
53i    543    555    568    580 
654    667    679    691    704 
777    79°    802    814     827 
900    913     925     937     949 

469    481     494    506    518 
593    603    617    630    642 
716    728     741     753    765 
839     851     864    876     888 
962    974    986    998  *oi  i 

i 

13 

I  3 

355 
356 
357 
358 
359 

55023    035    047    060    072 
145     157     169     182     194 
267     279     291     303     315 
388    400    413    425    437 
509    522    534    546    558 

084    096     108     121     133 

206     218     230    242     253 

328   340  352  364  376 

449    461     473    485    497 
570    582    594    606    618 

2 

3 

4 

7 

2/6 
3/9 

5/2 

7,'8 
9/1 

360 

361 
362 
363 
364 

630    642    654    666    678 

75i     763    775    787    799 
871     883     895    907     919 
991  *003  *ois  *027  3,038 

56  IIO      122      134      146      158 

691    703    713    727    739 
8n     823     833     847     859 

93i    943    953    967    979 
#050  #062  #074  $086  #098 
170     182     194    205     217 

8 
9 

io/4 
n/7 

365 
366 
367 
368 
369 

229      241      253      265      277 
348      360      372      384      396 
467      478      490      502      514 

58?    597    608    620    632 
703    714    726    738    750 

289    301     312    324    336 
407    419    431     443    453 
526    538    549    561    573 
644    656    667    679    691 
701     773    783    797     808 

i 

2 

3 
4 

12 

1,2 
2,4 

3,6 
4/8 

370 

37i 
372 
373 
374 

820    832    844    855    867 
937    949    961     972    984 
57  054    066    078    089     101 
171     183     194    206    217 
287    299    310    322    334 

879     891     902    914    926 
996  *oo8  #019  *03  i  #043 
113     124     136     148     159 
229     241     252    264     276 
345    357    368    380    392 

7 
8 

9 

6,0 

7/2 

8,4 
9/6 
10,8 

P 

3 

379 

403    413    426    438    449 

519    530    542    553    563 
634    646    657    669    680 
749    76i     772    784    795 
864     875     887     898    910 

461    473    484    496    507 
576    588     600    6n     623 
692    703    713    726    738 
807     818     830    841     852 
921    933    944    955    967 

i 

2 

11 

i  i 

380 

38i 
382 
383 
384 

978    990  #ooi  #013  #024 
58  092    104    11=;     127     138 
206    218    229    240    252 
320    331    343    354    365 
433    444    456    4^7    478 

*03$  #047  #058  #070  *o8i 
149     161     172     184     193 
263     274     286     297     309 
377     388     399    4io    422 
490    501    512    524    533 

3 
4 

33 

44 

385 

386 

387 
388 

389 

54^    557    5*>9    580    591 
659    670    68  i    692    704 
771     782    794    803     816 
883     894     906    917    928 
993  *oo6  *oi7  #028  5*040 

602    614    623    636    647 
713    726    737     749    760 
827     838     850    86  i     872 
939    950    961     973     984 
*05  i  *o62  *073  ^084  ^09$ 

9 

99 
i  n 

390 

39i 
392 
393 
394 

59  106    118     129    140    151 
218    229    240    251     262 
329    34°    35i    362    373 
439    450    461    472    483 
550    S^i    572    583    594 

162     173     184     195     207 
273     284     295     306    318 
384    395     406    417    428 
494    506     517     528     539 
603    616    627     638     649 

2 

3 
4 
5 

1U 

1,0 
2,0 

3/o 
4/o 
5<° 

395 
396 

399 

660    671    682    693    704 
770    780    791    802    813 
879    890    901    912    923 
988     999  *oio  *02i  $032 
60097     108     119     130     141 

713    726    737    748    759 
824     835     846    857     868 
934     943    956    966    977 
*043  *054  *o63  ^076  *o86 
152     163     173     184     195 

6 

7 
8 

9 

6,0 
7/o 
8,0 
9/o 

400 

206     217     228     239     249 

260    271     282     293    304 

N. 

L.   0       1       2       3       4 

56789 

I 

>.  P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

400 

401 
402 

403 
404 

60  206  217  228  239  249 

3H  32^  336  347  358 
423  433  444  455  466 

531  541  552  563  574 
638  649  660  670  68  i 

260  271  282  293  304 
369  379  390  401  412 
477  487  498  509  520 
584  595  606  617  627 
692  703  713  724  735 

i 

2 

3 
4 

7 
8 

9 

i 

2 

3 
4 

7 
8 

9 

i 

2 

0 

4 
5 
o 

8 
9 

LI 

1,1 

2,2 

3,3 
4,4 

ft 

7/7 

8,8 

9/9 
10 

1,0 
2/O 

3,o 
4/o 
5/° 
6,0 
7,o 
8,0 
9,o 

9 

o,9 
1,8 

8 

4,5 
5,4 
6,3 

B 

405 
406 
407 
408 
409 

746  756  767  778  788 
853  863  874  885  895 
959  970  981  991  #002 
61  066  077  087  098  109 

172   183   194   204   2I§ 

799  810  821  831  842 
906  917  927  938  949 
#013  #023  #034  #045  #055 
119  130  140  151  162 
225  236  247  257  268 

410 

411 
412 

4i3 

414 

278  289  300  310  321 
384  395  405  416  426 
490  500  511  521  532 
595  606  616  627  637 
700  711  721  731  742 

331  342  352  363  374 

437  448  458  469  479 

542  553  563  574  584 
648  658  669  679  690 

752  763  773  784  794 

4i5 
416 

417 
418 
419 

805  815  826  836  847 
909  920  930  941  951 
62  014  024  034  045  055 
118  128  138  149  159 

221   232   242   252   263 

857  868  878  888  899 
962  972  982  993  #003 
066  076  086  097  107 

170   ISO   190   201   211 
273   284   294   304   315 

420 

421 

422 

423 
424 

325  33$  346  356  366 
428  439  449  459  469 
531  542  552  562  572 
634  644  655  665  675 
737  747  757  7&7  778 

377  387  397  4°8  418 
480  490  500  511  521 
583  593  603  613  624 
685  696  706  716  726 
788  798  808  818  829 

42! 
426 

428 
429 

839  849  859  870  880 
941  951  961  972  982 
63043  053  063  073  083 
144  155  165  175  185 
246  256  266  276  286 

890  900  910  921  931 

992  #002  #012  #022  #033 
094   104   114   124   134 
195   205   215   225   236 

296  306  317  327  337 

430 

43i 
432 
433 
434 

347  357  367  377  387 
448  458  468  478  488 
548  558  568  579  589 
649  659  669  679  689 
749  759  769  779  7»9 

397  407  417  428  438 
498  508  518  528  538 
599  609  619  629  639 
699  709  719  729  739 
799  809  819  829  839 

435 
436 
437 
438 
439 

849  859  869  879  889 
949  959  969  979  988 
64  048  058  068  078  088 
147  157  167  177  187 
246  256  266  276  286 

899  909  919  929  939 
998  #008  #01  8  #028  #038 
098  108  118  128  137 
197  207  217  227  237 
296  306  316  326  335 

440 

441 
442 
443 
444 

345  355  365  375  3^ 
444  454  464  473  4^3 
542  552  562  572  582 
640  650  660  670  680 
738  748  758  768  777 

395  404  414  424  434 

493  503  513  523  532 
591  601  6ri  621  631 
689  699  709  719  729 
787  797  807  816  826 

445 
446 

447 
448 

449 

836  846  856  86$  875 
933  943  953  963  972 
65  031  040  050  060  070 
128  137  147  157  167 
225  234  244  254  263 

885  895  904  914  924 
982  992  #002  #011  #021 
079  089  099  108  118 
176  186  196  205  215 
273  283  292  302  312 

450 

321  331  341  350  360 

369  379  389  398  408 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P 

.P. 

450 

45i 

452 
453 
454 

65321  331  341  350  360 
418  427  437  447  456 
514  523  533  543  552 
610  619  629  639  648 
706  715  725  734  744 

369  379  389  398  408 
466  475  485  495  504 
562  571  581  591  600 
658  667  677  686  696 
753  763  772  782  792 

i 

2 

3 

4 
5 
6 

7 
8 

9 

2 

3 
4 

I 

9 

i 

2 

3 
4 
5 
6 

I 

9 

10 

1,0 

2/O 

3/° 
4/o 
5/° 
6,0 
7/o 
8,0 
9/o 

9 

o,9 
1,8 

2,7 

3/6 
4/5 
5/4 
6/3 

7/2 

8/1 

8 

0/8 
1,6 

2/4 

3/2 
4/o 
4/8 
5/6 
6/4 

7/2 

455 
456 
457 
458 
459 

801  8  I  i  820  830  839 
896  906  916  925  935 
992  #ooi  #011  #020  #030 
66  087  096  106  115  124 
181  191  200  210  219 

849  858  868  877  887 
944  954  963  973  982 
#039  #049  #058  #068  #077 
134  143  153  162  172 
229  238  247  257  266 

460 

461 

462 

463 
464 

276  285  295  304  314 
370  380  389  398  408 
464  474  483  492  502 
558  567  577  586  596 
652  661  671  680  689 

323  332  342  351  36i 
417  427  436  445  455 
511  521  530  539  549 
605  614  624  633  642 
699  7o8  717  727  736 

4^ 
466 

467 
468 
469 

745  755  764  773  783 
839  848  857  867  876 
932  941  950  960  969 
67  025  034  043  052  062 
117  127  136  145  154 

792  801  811  820  829 
885  894  904  913  922 
978  987  997  #006  #015 
071  080  089  099  108 

164   173   182   191   201 

470 

47i 
472 
473 
474 

210   219   228   237   247 

302  311  321  330  339 
394  403  413  4*2  431 
486  495  504  514  523 
578  587  596  605  614 

256  265  274  284  293 
348  357  367  376  385 
440  449  459  468  477 
532  541  550  560  569 
624  633  642  651  660 

475 
476 

477 
478 
479 

669  679  688  697  706 
761  770  779  788  797 
852  861  870  879  888 
943  952  961  970  979 
68  034  043  052  061  070 

715  724  733  742  752 
806  81^  825  834  843 
897  906  916  925  934 
988  997  #006  #oic;  ^024 
079  088  097  1  06  115 

480 

481 
482 
483 
484 

124  133  142  151  160 
215  224  233  242  251 
305  3J4  323  332  34i 
395  404  413  422  431 
485  494  502  511  520 

169  178  187  196  205 
260  269  278  287  296 
35o  359  368  377  386 
440  449  458  467  476 
529  538  547  556  565 

485 
486 
487 
488 
489 

574  583  592  601  610 
664  673  68  i  690  699 
753  762  771  780  789 
842  851  860  869  878 
931  940  949  958  966 

619  628  637  646  655 
708  717  726  735  744 
797  806  815  824  833 
886  89^  904  913  922 
975  984  993  *oo2  *on 

490 

491 
492 
493 
494 

69  020  028  037  046  05=; 
108  117  126  135  144 
197  205  214  223  232 
285  294  302  311  320 
373  381  390  399  408 

064  073  082  090  099 
152  161  170  179  188 
241  249  258  267  276 
329  338  346  355  364 
417  425  434  443  452 

495 
496 

497 
498 
499 

461  469  478  487  496 
548  557  566  574  583 
636  644  653  662  671 
723  732  740  749  758 
8  10  819  827  836  845 

504  513  522  531  539 
592  601  609  618  627 
679  688  697  705  714 
767  775  784  793  801 
854  862  871  880  888 

500 

897  906  914  923  932 

940  949  958  966  975 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

I 

'.P. 

500 

501 
502 

5°3 
504 

69  897  906  914  923  932 
984  992  #ooi  *oio  *oi8 
70  070  079  088  096  105 
157  165  174  183  191 
243  252  260  269  278 

940  949  958  966  975 
*027  ^036  *044  *053  *o62 
114  122  131  140  148 

200   209   217   226   234 

286  295  303  312  321 

505 
506 

5°7 
508 

509 

329  338  346  353  364 
415  424  432  441  449 
501  509  518  526  535 
586  595  603  612  621 
672  680  689  697  706 

372  381  389  398  406 

458  467  475  484  492 
544  552  561  569  578 
629  638  646  655  663 
714  723  731  740  749 

i 

2 

3 

9 

o/9 
1,8 

2.7 

510 

5n 
512 
5i3 
5i4 

757  766  774  783  791 
842  851  859  868  876 
927  935  944  952  961 

71  OI2   O2O   029   037   046 
096   lO^   113   122   130 

800  808  817  825  834 
885  893  902  910  919 
969  978  986  995  *oo3 
054  063  071  079  088 
139  147  155  164  172 

4 

7 
8 

9 

36 
4,5 
5,4 
6,3 
7.2 
8/1 

5i5 
5i6 
5i7 
518 

5i9 

181  189  198  206  214 

265   273   282   29O   299 

349  357  366  374  383 
433  441  450  458  466 
517  525  533  542  550 

223  231  240  248  257 
307  3Jj  324  S32  34i 
391  399  408  416  425 
.  473  483  492  3oo  508 
559  567  575  584  592 

520 

521 

522 

523 
524 

600  609  617  625;  634 
684  692  700  709  717 
767  775  784  792  800 
850  858  867  875  883 
933  94i  950  958  966 

642  650  659  667  675 

725  734  742  750  759 
809  817  825  834  842 
892  900  908  917  925 
973  983  99i  999  *oo8 

i 

2 

3 

8 

0,8 
1,6 

2,4 

52I 
526 

527 

528 

529 

72  016  024  032  041  049 
099  107  115  123  132 
181  189  198  206  214 
263  272  280  288  296 
346  354  362  370  378 

057  066  074  082  090 
140  148  156  165  173 

222   230   239   247   255 

304  313  321  329  337 
387  393  403  4"  419 

4 

7 
8 

9 

3,2 
4,0 
4,8 
5,6 
6,4 

7/2 

530 

53i 
532 
533 
534 

428  436  444  452  460 
509  518  526  534  542 
591  599  607  616  624 
673  68  i  689  697  705 
754  762  770  779  787 

469  477  483  493  501 

550  558  567  573  583 
632  640  648  656  665 
713  722  730  738  746 
795  803  811  819  827 

537 
538 
539 

835  843  852  860  868 
916  925  933  941  949 
997  #006  #014  *022  #030 
73  078  086  094  102  in 
159  167  175  183  191 

876  884  892  900  908 

957  965  973  98i  989 
#038  ^046  *054  *o62  *070 
119  127  135  143  151 
199  207  215  223  231 

i 

2 

3 

7 

o,7 
i,4 

2,1 

540 

54i 
542 
543 
544 

239  247  255  263  272 
320  328  336  344  352 
400  408  416  424  432 
480  488  496  504  512 
560  568  576  584  592 

280  288  296  304  312 
360  368  376  384  392 
440  448  456  464  472 
520  528  536  544  552 
600  608  616  624  632 

4 

7 
8 

9 

2,8 
3,5 

4,2 

4,9 
5,6 
6,3 

545 
546 
547 
548 
549 

640  648  656  664  672 
719  727  73$  743  751 
799  807  815  823  830 
878  886  894  902  910 
957  965  973  981  989 

679  687  695  703  711 

759  767  773  783  79i 
838  846  854  862  870 
918  926  933  941  949 
997  *oo5  *oi3  *020  *028 

550 

74  036  044  052  060  068 

076  084  092  099  107 

N. 

L.  0   1   2   3   4 

56789 

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N. 

L.  0   1   2   34 

56789 

P.P. 

550 

55i 
552 
553 

554 

74  036  044  052  060  068 
115  123  131  139  147 

194   202   210   2l8   225 
273   280   288   296   304 

35i  359  367  374  382 

076  084  092  099  107 
155  162  170  178  186 
233  241  249  257  265 
312  320  327  335  343 
390  398  406  414  421 

8 

i  0,8 

2  1,6 

3  2,4 
4  S,2 
5  4,o 
6  4,8 
7  5,6 
8  6,4 
9  7,2 

7 

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2  1,4 

321 
4  2,8 
5  3,5 
6  4,2 
7  4,9 
8  5,6 
9  6,3 

555 
556 
557 
558 
559 

429  437  445  453  461 
507  5i5  523  53i  539 
586  593  601  609  617 
663  671  679  687  695 
74i  749  757  764  772 

468  476  484  492  500 
547  554  562  570  578 
624  632  640  648  656 
702  710  718  726  733 
780  788  796  803  811 

560 

56i 

562 

563 
564 

819  827  834  842  850 
896  904  912  920  927 
974  981  989  997  *oo5 
75  051  059  066  074  082 
128  136  143  151  159 

858  865  873  881  889 
935  943  950  958  966 

*OI2  *020  *028  *03S  *043 
089  097  IO5  H3  I2O 

166  174  182  189  197 

5S 
566 

567 
568 

569 

203   213   220   228   236 
282   289   297   305   312 

358  366  374  381  389 
435  442  450  458  465 
511  519  526  534  542 

243  251  259  266  274 
320  328  335  343  351 
397  404  412  420  427 
473  481  488  496  504 
549  557  565  572  580 

570 

57i 
572 
573 
574 

587  595  603  610  618 
664  671  679  686  694 
740  747  755  762  770 
815  823  831  838  846 
891  899  906  914  921 

626  633  641  648  656 
702  709  717  724  732 
778  785  793  800  808 
853  861  868  876  884 
929  937  944  952  959 

II 

577 
578 
579 

967  974  982  989  997 
76  042  050  057  065  072 
118  125  133  140  148 
193  200  208  215  223 
268  275  283  290  298 

*oo5  *oi2  *02o  *027  $035 
080  087  095  103  no 
155  163  170  178  185 

230  238  243  253  260 
305  313  320  328  335 

580 

581 
582 

583 
584 

343  350  358  365  373 
418  425  433  440  448 
492  500  507  515  522 
567  574  582  589  597 
641  649  656  664  671 

380  388  395  403  410 
455  462  470  477  485 

530  537  545  552  559 
604  612  619  626  634 
678  686  693  701  708 

585 
586 

587 
588 
589 

716  723  730  738  745 
790  797  805  812  819 
864  871  879  886  893 
938  945  953  96o  967 
77  012  019  026  034  041 

753  760  768  775  782 
827  834  842  849  856 
901  908  916  923  930 
975  982  989  997  *oo4 
048  056  063  070  078 

590 

59i 
592 
593 
594 

085  093  ioo  107  115 
159  166  173  181.  188 
232  240  247  254  262 
305  313  320  327  335 
379  386  393  401  408 

122  129  137  144  151 
195  203  210  217  225 
269  276  283  2QI  298 

342  349  357  364  37i 
415  422  430  437  444 

595 
596 
597 
598 
599 

452  459  466  474  481 

525  532  539  546  554 
597  605  612  619  627 
670  677  685  692  699 
743  75o  757  764  772 

488  495  503  510  517 
561  568  576  583  590 
634  641  648  656  663 
706  714  721  728  735 
779  786  793  801  808 

600 

815  822  830  837  844 

851  859  866  873  880 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

600 

601 
602 

S3 
604 

77815  822  830  837  844 
887  895  902  909  916 
960  967  974  981  988 
78032  039  046  053  061 
104  in  118  125  132 

851  859  866  873  880 

924  93i  938  945  952 
996  *oo3  *oio  #017  #025 
068  075  082  089  097 
140  147  154  161  168 

i 

2 

3 
4 

I 

7 
8 

9 

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2 

3 
4 
5 
6 

9 

i 

2 

3 
4 

9 

8 

0/8 
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2,4 

3,2 

4/o 
4,8 

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7,2 

7 

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2,1 

2,8 
3/5 

4,2 

n 

6,3 

6 

0,6 

1,2 

1,8 

2,4 

3'°. 

3,6 
4,2 
4,8 
5,4 

607 
608 
609 

176  183  190  197  204 
247  254  262  269  276 
319  326  333  340  347 
390  398  405  412  419 
462  469  476  483  490 

211  219  226  233  240 

283  290  297  305  312 
355  362  369  376  383 
426  433  440  447  455 
497  504  512  519  526 

610 

611 
612 
613 
614 

533  540  547  554  561 
604  611  618  625  633 
67^  682  689  696  704 
746  753  760  767  774 
817  824  831  838  845 

569  576  583  590  597 
640  647  654  66  i  668 
711  718  725  732  739 
781  789  796  803  8  10 
852  859  866  873  880 

615 
616 
617 
618 
619 

888  895  902  909  916 
958  965  972  979  986 
79029  036  043  050  057 
099  106  113  120  127 
169  176  183  190  197 

923  930  937  944  951 
993  #000  *oo7  #014  *o2i 
064  071  078  085  092 
134  141  148  155  162 
204  211  218  225  232 

620 

621 
622 
623 
624 

239  246  253  260  267 
309  316  323  330  337 
379  386  393  400  407 
449  456  463  470  477 
5i8  525  532  539  546 

274  281  288  295  302 
344  35i  358  365  372 
414  421  428  435  442 
484  491  498  505  511 
553  56o  567  574  58i 

625 

626 
627 
628 
629 

588  595  602  609  616 
657  664  671  678  685 
727  734  741  748  754 
796  803  810  817  824 
865  872  879  886  893 

623  630  637  644  650 
692  699  706  713  720 
761  768  775  782  789 
831  837  844  851  858 
900  906  913  920  927 

630 

631 
632 
633 
634 

934  941  948  955  962 
80  003  oio  017  024  030 
072  079  085  092  099 
140  147  154  161  168 
209  216  223  229  236 

969  975  982  989  996 
037  044  051  058  065 
106  113  120  127  134 
175  182  188  195  202 

243  250  257  264  271 

635 
636 

637 
638 

639 

277  284  291  298  305 
346  353  359  366  373 
414  421  428  434  441 
482  489  496  502  509 
550  557  564  570  577 

312  318  325  332  339 
380  387  393  400  407 
448  455  462  468  475 
5i6  523  530  536  543 
584  591  598  604  611 

640 

641 

642 
643 
644 

618  625  632  638  645 
686  693  699  706  713 
754  760  767  774  781 
821  828  835  841  848 
889  895  902  909  916 

652  659  665  672  679 
720  726  733  740  747 
787  794  801  808  814 
855  862  868  875  882 
922  929  936  943  949 

645 
646 
647 
648 
649 

956  963  969  976  983 
8  1  .023  030  037  043  050 
090  097  104  in  117 
158  164  171  178  184 
224  231  238  245  251 

990  996  *oo3  *oio  *oi7 
057  064  070  077  084 
124  131  137  *44  151 

igi   198   204   211   2l8 
258   265   271   278   285 

650 

291  298  305  311  318 

325  331  338  345  351 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

650 

65i 
652 

653 
654 

81  291  298  305  311  318 
358  365  37i  378  385 
425  431  438  445  451 
491  498  505  511  518 
558  564  57i  578  584 

325  33i  338  345  35i 
391  398  405  411  418 

458  465  47i  478  485 
525  53i  538  544  55i 
591  598  604  6n  617 

7 

I  0,7 
2  1,4 
3  2,1 
4  2,8 
5  3,5 
6  4,2 

7  4,9 
8  5,6 
9  6,3 

6 

i  0,6 

2  1,2 

3  1,8 
4  2,4 

1  3'2 
6  3,6 

7  4,2 
8  4,8 

9  5,4 

655 
656 

657 
658 

659 

624  631  637  644  651 
690  697  704  710  717 
757  763  770  776  783 
823  829  836  842  849 
889  895  902  908  915 

657  664  671  677  684 

723  73°  737  743  75° 
790  796  803  809  816 
856  862  869  875  882 
921  928  935  941  948 

660 

661 
662 
663 
664 

954  961  968  974  981 

82  020   027   033   040   046 
086   092   099   105   112 
151   158   164   171   178 
217   223   230   236   243 

987  994  *ooo  *oo7  *oi4 
053  060  066  073  079 
119  125  132  138  145 
184  191  197  204  210 
249  256  263  269  276 

667 
668 
669 

282   289   295   302   308 

347  354  360  367  373 
413  419  426  432  439 
478  484  491  497  504 
543  549  556  562  569 

315  321  328  334  341 
380  387  393  400  406 
445  452  458  465  471 
510  517  523  530  536 
575  582*  588  595  601 

670 

671 
672 

S3 

674 

607  614  620  627  633 

672  679  68^  692  698 

737  743  75o  756  763 
802  808  814  821  827 
866  872  879  885  892 

640  646  653  659  666 
705  711  718  724  730 
769  776  782  789  795 
834  840  847  853  860 
898  905  911  918  924 

675 
676 
677 
678 
679 

930  937  943  95o  956 
995  *ooi  *oo8  #014  #020 
83  059  065  072  078  085 
123  129  136  142  149 
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963  969  975  982  988 
*027  ^033  #040  ^046  ^052 
091  097  104  no  117 
155  161  168  174  181 
219  225  232  238  245 

680 

68  1 
682 
683 
684 

251  257  264  270  276 
315  321  327  334  340 
378  385  39i  398  404 
442  448  455  461  467 
506  512  518  525  53I 

283  289  296  302  308 
347  353  359  366  372 
410  417  423  429  436 
474  480  487  493  499 
537  544  550  556  563 

685 
686 
687 
688 
689 

569  575  582  588  594 
632  639  645  651  658 
696  702  708  715  721 
759  765  77i  778  784 
822  828  835  841  847 

601  607  613  620  626 

664  670  677  683  689 
727  734  740  746  753 
790  797  803  809  816 
853  860  866  872  879 

690 

691 
692 
693 
694 

885  891  897  904  910 
948  954  960  967  973 
84  on  017  023  029  036 
073  080  086  092  098 
136  142  148  155  161 

916  923  929  935  942 
979  985  992  998  *oo4 
042  048  055  06  I  067 
105  in  117  123  130 
J67  173  180  186  192 

695 
696 
697 
698 
699 

198   205   211   217   223 
26l   267   273   280   286 
323   330   336   342   348 
386   392   398   404   410 

448  454  460  466  473 

230  236  242  248  255 
292  298  305  311  317 

354  361  367  373  379 
417  423  429  435  442 
479  485  491  497  504 

700 

510  516  522  528  535 

54i  547  553  559  566 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56   7   89 

P.P. 

700 

701 
702 

703 

704 

84510  516  522  528  535 
572  578  584  590  597 
634  640  646  652  658 
696  702  708  714  720 
757  763  770  776  782 

54i  547  553  559  566 
603  609  615  621  628 
665  671  677  683  689 
726  733  739  745  751 
788  794  800  807  813 

2 

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3 
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3 
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7 
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4,9 
5,6 
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0,6 

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5 

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705 
706 
707 
708 
709 

819  82^  831  837  844 
880  887  893  899  905 
942  948  954  960  967 
85  003  009  016  022  028 
065  071  077  083  089 

850  856  862  868  874 
911  917  924  930  936 

973  979  985  99i  997 
034  040  046  052  058 

095   101   107   114   120 

710 

711 
712 
713 
714 

126  132  138  144  150 

187   193   199   20f;   211 
248   254   260   266   272 

309  31^  321  327  333 
370  376  382  388  394 

156  163  169  175  181 

217  224  230  236  242 
278  285  291  297  303 

339  345  352  358  364 
400  406  412  418  425 

715 
716 
717 
718 
719 

43i  437  443  449  455 
491  497  503  509  516 
552  558  564  570  576 
612  6i3  625  631  637 
673  679  685  691  697 

461  467  473  479  48$ 
522  528  534  540  546 
582  588  594  600  606 
643  649  655  661  667 
703  709  715  721  727 

720 

721 
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723 
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733  739  745  75*  757 
794  800  806  812  818 
854  860  866  872  878 
914  920  926  932  938 
974  980  986  992  998 

763  769  775  781  788 
824  830  836  842  848 
884  890  896  902  908 
944  950  956  962  968 

#<X>4  #OIO  *Ol6  *O22  #O28 

725 
726 

727 
728 
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86  034  040  046  052  058 
094  loo  106  112  118 
153  159  165  171  177 
213  219  223  231  237 
273  279  285  291  297 

064   070   076   082   O88 
124   130   136   14!   147 
183   189   19^   2OI   2O7 
243   249   255   26l   267 
303   308   314   320   326 

730 

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332  338  344  350  356 
392  398  404  410  41^ 
451  457  463  469  475 
510  516  522  528  534 
570  576  581  587  593 

362  368  374  380  386 
421  427  433  439  445 
481  487  493  499  504 
540  546  552  558  564 
599  6oc;  611  617  623 

735 
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629  635  641  646  652 
688  694  700  705  711 

.  747  753  759  764  77O 
806  812  817  823  829 
864  870  876  882  888 

658  664  670  676  682 
717  723  729  735  741 
776  782  788  794  800 
835  841  847  853  859 
894  900  906  911  917 

740 

74i 
742 
743 
744 

923  929  935  941  947 
982  988  994  999  *oo$ 
87  040  046  052  058  064 

099   105   III   Il6   122 

157  163  169  175  181 

953  958  964  970  976 
*on  #017  #023  *029  *035 
070  07=;  08  1  087  093 
128  134  140  146  151 
186  192  198  204  210 

745 
746 
747 
748 

749 

2l6   221   227   233   239 
274   280   286   291   297 

332  338  344  349  355 
390  396  402  408  413 
448  454  460  466  471 

245  251  256  262  268 
303  309  315  320  326 
361  367  373  379  384 
419  425  431  437  442 
477  483  489  495  500 

750 

506  512  518  523  529 

535  54i  547  552  558 

N. 

L.  0   1   2   3   4 

56789 

P 

.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

750 

75i 
752 
753 
754 

87  506  512  518  523  529 
564  570  576  581  587 
622  628  633  639  645 
679  685  691  697  703 
737  743  749  754  760 

535  54i  547  552  558 
593  599  6°4  610  616 
651  656  662  668  674 
708  714  720  726  731 
766  772  777  783  789 

6 

i  0,6 

2  1,2 

3  1,8 
4  2,4 

i!:? 

U:s2 

9  5/4 

5 

i|°,5 

2  1,0 

3  i,5 
4  2,0 
5  2,5 
6  3.o 
7  3,5 
8  4,0 

9  4,5 

755 
756 
757 
758 
759 

795  800  806  812  818 
852  858  864  869  875 
910  915  921  927  933 
967  973  978  984  990 
88  024  030  036  041  047 

823  829  835  841  846 
88  i  887  892  898  904 
938  944  950  955  961 
996  *ooi  $007  $013  *oi8 
°53  °58  064  070  076 

760 

761 
762 

763 

764 

08  i  087  093  098  104 
138  144  150  156  161 
195  201  207  213  218 
252  258  264  270  275 
309  315  321  326  332 

no  116  121  127  133 
167  173  178  184  190 
224  230  23^  241  247 
281  287  292  298  304 
338  343  349  353  360 

765 
766 
767 
768 
769 

366  372  377  383  389 
423  429  434  440  446 
480  485  491  497  502 
536  542  547  553  559 
593  598  604  610  615 

395  400  406  412  417 
451  457  463  468  474 
508  513  519  525  530 
564  570  576  581  587 
621  627  632  638  643 

770 

77i 
772 
773 
774 

649  655  660  666  672 
705  711  717  722  728 
762  767  773  779  784 
818  824  829  835  840 
874  880  885  891  897 

677  683  689  694  700 
734  739  745  75°  756 
79°  795  801  807  812 
846  852  857  863  868 
902  908  913  919  925 

775 
776 

777 
778 

779 

930  936  941  947  953 
986  992  997  ^003  *oc>9 
89  042  048  053  059  064 
098  104  109  115  i  20 
154  159  165  170  176 

958  964  969  975  981 
*oi4  *020  *025  *03  i  *037 
070  076  08  i  087  092 
126  131  137  143  148 
182  187  193  198  204 

780 

781 
782 

783 
784 

2O9   21$   221   226   232 

265   271   276   282   287 

321  326  332  337  343 
376  382  387  393  398 
432  437  443  448  454 

237  243  248  254  260 
293  298  304  310  315 
348  354  360  365  371 
404  409  415  421  426 
459  465  470  476  481 

785 
786 

787 
788 
789 

487  492  498  504  509 
542  548  553  559  564 
597  603  609  614  620 
653  658  664  669  675 
708  713  719  724  730 

515  520  526  531  537 
570  575  58i  586  592 
625  631  636  642  647 
680  686  691  697  702 
735  74i  746  752  757 

790 

791 
792 
793 
794 

763  768  774  779  785 
818  823  829  834  840 
873  878  883  889  894 

927  933  938  944  949 
982  988  993  998  *004 

790  796  801  807  812 
845  851  856  862  867 
900  905  911  916  922 
953  96o  966^971  977 

*009  *OI3  *020  *026  *03  I 

795 
796 

797 
798 

799 

90037  042  048  053  059 
091  097  102  108  113 
146  151  157  162  168 

200   206   211   217   222 

255  260  266  271  276 

064  069  075  080  086 
119  124  129  135  140 
173  179  184  189  195 
227  233  238  244  249 
282  287  293  298  304 

800 

309  314  320  325  331 

336  342  347  352  358 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

800 

801 
802 
803 
804 

90309  314  320  325  331 
363  369  374  380  385 
417  423  428  434  439 
472  477  482  488  493 
526  531  536  542  547 

336  342  347  352  358 
390  396  401  407  412 
445  450  455  461  466 
499  5°4  5°9  5i5  520 
553  558  563  569  574 

6 

i  0,6 

2  1,2 

3  1,8 
4  2,4 

I  3'°. 
6  3,6 

7  4,2 
8  4,8 
9  5/4 

5 

i  o,5 

2  1,0 

3  1,5 
4  2,0 
5  2,5 
6  3,° 
7  3,5 
8  4,0 

9  4,5 

805 
806 
807 
808 
809 

580  585  590  596  601 
634  639  644  650  655 
687  693  698  703  709 
741  747  752  757  763 
795  800  806  811  816 

607  612  617  623  628 
660  666  671  677  682 
714  720  725  730  736 
768  773  779  784  789 
822  827  832  838  843 

810 
BIX 

812 

813 
814 

849  854  859  865  870 
902  907  913  918  924 
956  961  966  972  977 
91  009  014  020  025  030 
062  068  073  078  084 

875  88  i  886  891  897 

929  934  940  945  950 
982  988  993  998  *oo4 
036  041  046  052  057 
089  094  loo  105  no 

815 
816 
817 
818 
819 

116  121  126  132  137 
169  174  i  80  185  190 

222   228   233   238   243 
275   28l   286   291   297 

328  334  339  344  35o 

142  148  153  158  164 

196  201  206  212  217 
249  254  259  265  270 
302  307  312  318  323 

355  360  365  371  376 

820 

821 
822 
823 
824 

381  387  392  397  403 
434  440  445  450  455 
487  492  498  503  508 
540  545  551  556  561 
593  598  603  609  614 

408  413  418  424  429 

461  466  471  477  482 
514  519  524  529  535 
566  572  577  582  587 
619  624  630  635  640 

825 
826 
827 
828 
829 

645  651  656  661  666 
698  703  709  714  719 
75i  756  76i  766  772 
803  808  814  819  824 
855  861  866  871  876 

672  677  682  687  693 
724  730  735  740  745 
777  782  787  793  798 
829  834  840  845  850 
882  887  892  897  903 

830 

831 
832 
833 
834 

908  913  918  924  929 
960  965  971  976  981 
92  012  018  023  028  033 
065  070  075  080  085 

117   122   127   132   137 

934  939  944  95<>  955 
986  991  997  *oo2  *oo7 
038  044  049  054  059 
091  096  101  106  in 
143  148  153  158  163 

835 
836 

837 
838 

839 

169   174   179   184   189 
221   226   231   236   241 
273   278   283   288   293 

324  33°  335  340  345 
376  381  387  392  397 

195  200  20=;  210  215 
247  252  257  262  267 

298  304  309  314  319 

350  355  301  366  37i 
402  407  412  418  423 

840 

841 
842 

843 
844 

428  433  438  443  449 
480  485  490  495  500 
53i  536  542  547  552 
583  588  593  598  603 
634  639  645  650  655 

454  459  464  469  474 
505  511  516  521  526 
557  562  567  572  578 
609  614  619  624  629 
660  665  670  675  681 

845 
846 

847 
848 
849 

686  691  696  701  706 
737  742  747  752  758 
788  793  799  804  809 
840  845  850  855  860 
891  896  901  906  911 

711  716  722  727  732 
763  768  773  778  783 
814  819  824  829  834 
865  870  875  88  i  886 
916  921  927  932  937 

850 

942  947  952  957  962 

967  973  978  983  988 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

850 

851 
852 
853 
854 

92942  947  952  957  962 
993  998  #003  *oo8  *oi3 
93044  049  054  059  064 
095  100  105  no  115 
146  151  156  161  166 

967  973  978  983  988 
*oi8  *024  #029  *034  #039 
069  075  080  085  090 

I2O   125   131   136   141 

171  176  181  186  192 

i 

2 

3 
4 

7 
8 

9 

i 

2 

3 
4 

i 

9 

i 

2 

3 
4 

7 
8 

9 

6 

0,6 

J,2 

1,8 

2,4 

3'2 

3,6 

4,2 

4,8 
5,4 

5 

o,5 
I,° 
i,5 
2,0 

2,5 

3,o 
3,5 
4,o 
4,5 

4 

o,4 
0,8 

1,2 

1,6 

2,0 
2,4 

2,8 

8 

855 
856 
857 
858 

859 

197  202  207  212  217 
247  252  258  263  268 
298  3°3  308  313  3l8 
349  354  359  3^4  3^9 
399  404  409  414  420 

222   227   232   237   242 

273  278  283  288  293 
323  328  334  '  339  344 

374  379  384  389  394 
425  430  435  440  445 

860 

861 
862 
863 

864 

450  455  460  465  470 
5oo  505  510  515  520 
551  556  561  566  571 
601  606  6n  616  621 
651  656  661  666  671 

475  480  485  490  495 
526  531  536  541  546 
576  581  586  591  596 
626  631  636  641  646 
676  682  687  692  697 

865 
866 
867 
868 
869 

702  707  712  717  722 

752  757  762  767  772 
802  807  812  817  822 
852  857  862  867  872 
902  907  912  917  922 

727  732  737  742  747 
777  782  787  792  797 
827  832  837  842  847 
877  882  887  892  897 
927  932  937  942  947 

870 

871 
872 

873 
874 

952  957  962  967  972 
94  002  007  012  017  022 
052  057  062  067  072 
101  106  in  116  121 
151  156  161  166  171 

977  982  987  992  997 
027  032  037  042  047 
077  082  086  091  096 
126  131  136  141  146 
176  181  186  191  196 

875 
876 
877 
878 
879 

2OI   2O6   211   2l6   221 

250  255  260  265  270 
300  305  310  315  320 
349  354  359  3^4  3^9 
399  404  409  414  419 

226  231  236  240  245 
275  280  285  290  295 
325  33°  335  34°  345 
374  379  384  389  394 
424  429  433  438  443 

880 

881 
882 
883 
884 

448  453  458  463  468 
498  503  5°7  512  517 
547  SS2  557  562  567 
596  601  606  611  616 
645  650  655  660  665 

473  478  483  488  493 
522  527  532  537  542 
571  576  581  586  591 
621  626  630  63$  640 
670  675  680  685  689 

885 
886 
887 
888 
889 

694  699  704  709  714 

743  748  753  758  763 
792  797  802  807  812 
841  846  851  856  86  i 
890  895  900  905  910 

719  724  729  734  738 

768  773  778  783  787 
817  822  827  832  836 
866  871  876  880  885 
915  919  924  929  934 

890 

891 
892 
893 
894 

939  944  949  954  959 
988  993  998  *oo2  *oo7 
95  036  041  046  051  056 
08=;  090  095  loo  105 
134  139  143  148  153 

963  968  973  978  983 

*OI2  #017  *022  *027  *03  2 

061  066  071  07^  080 

109   114   119   124   129 

158  163  168  173  177 

895 
896 

897 
898 
899 

182   187   192   197   202 
231   236   240   245   250 
279   284   289   294   299 

328  332  337  342  347 
376  381  386  390  395 

2O7   211   2l6   221   226 
255   260   265   270   274 
303   308   313   318   323 

SS2  357  361  366  371 
400  405  410  415  419 

900 

424  429  434  439  444 

448  453  458  463  468 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

5   67   8   9 

P.P. 

900 

901 
902 

903 
904 

95424  429  434  439  444 
472  477  482  487  492 

S2!  525  530  535  540 
569  574  578  583  588 
617  622  626  631  636 

448  453  458  463  468 
497  501  506  511  516 
545  550  554  559  564 
593  598  602  607  612 
641  646  650  655  660 

5 

I!°/5 

2  1,0 

3  i/5 
4  2,0 
5  2,5 
6  3/o 
7  3/5 
8  4,0 

9  4/5 

4 

i  0,4 

2  0,8 

3  i/2 
4  1,6 
5  2/o 
6  2,4 
7  2,8 
83/2 
9  3/6 

905 
906 
907 
908 
909 

665  670  674  679  684 
713  718  722  727  732 
761  766  770  775  780 
809  813  818  823  828 
856  861  866  871  875 

689  694  698  703  708 
737  742  746  75i  756 
785  789  794  799  804 
832  837  842  847  852 
880  885  890  895  899 

910 

911 
912 

913 
914 

904  909  914  918  923 
952  957  961  966  971 
999  #004  *oo9  #014  *oig 
96  047  052  057  06  1  066 
095  099  104  109  114 

928  933  938  942  947 
976  980  985  990  995 
#023  *028  *033  ^.038  3.042 
071  076  080  085  090 
118  123  128  133  137 

9i5 
916 
917 
918 
919 

142  147  152  156  161 
190  194  199  204  209 
237  242  246  251  256 
284  289  294  298  303 
332  336  34i  346  350 

166  171  175  180  185 
213  218  223  227  232 
261  265  270  275  280 
308  313  317  322  327 
355  36o  365  369  374 

920 

921 
922 

923 
924 

379  384  388  393  398 
^426  431  435  440  445 
473  478  483  487  492 
520  525  530  534  539 
567  572  577  58i  586 

402  407  412  417  421 
450  454  459  464  468 
497  501  506  511  515 
544  548  553  55s  562 
591  595  600  605  609 

925 
926 

927 

928 

929 

614  619  624  628  633 

661  666  670  675  680 
708  713  717  722  727 

755  759  764  769  774 
802  806  811  816  820 

638  642  647  652  656 
685  689  694  699  703 
731  736  741  745  750 
778  783  788  792  797 
823  830  834  839  844 

930 

93i 
932 
933 
934 

848  853  858  862  867 
895  900  904  909  914 
942  946  951  956  960 
988  993  997  *oo2  ^007 
97035  °39  044  049  053 

872  876  881  886  890 
918  923  928  932  937 

963  970  974  979  984 
*oi  i  #016  #021  #025  3,030 
058  063  067  072  077 

935 
936 
937 
938 
939 

08  i  086  090  095  100 
128  132  137  142  146 
174  179  183  188  192 

220   225   230   234   239 

267  271  276  280  285 

104  109  114  ii  8  123 
I5I  r55  X6o  163  169 
197  202  206  211  216 
243  248  253  257  262 
290  294  299  304  308 

940 

941 
942 
943 
944 

313  317  322  327  331 

359  364  368  373  377 
405  410  414  419  424 
451  456  460  465  470 
497  502  506  511  516 

336  340  345  35o  354 
382  387  391  396  400 
428  433  437  442  447 
474  479  483  488  493 
520  523  529  534  539 

945 

946 

947 
948 
949 

543  548  552  557  562 
589  594  598  603  607 
635  640  644  649  653 
68  i  685  690  695  699 
727  731  736  740  745 

566  571  575  580  583 
612  617  621  626  630 
658  663  667  672  676 
704  708  713  717  722 
749  754  759  763  768 

950 

772  777  782  786  791 

795  800  804  809  813 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

950 

95i 
952 
953 
954 

97772  777  782  786  791 
818  823  827  832  836 
864  868  873  877  882 
909  914  918  923  928 
955  959  964  968  973 

795  800  804  809  813 
841  845  850  855  859 
886  891  896  900  905 
932  937  94i  946  950 
978  982  987  991  996 

5 

io/5 
2!  1,0 

3  |i/S 
4  2,o 

II2'5 
63/0 

7  3/5 
8  4,0 

9  4/5 

4 

I  0,4 

2  0,8 

3  1/2 
4  1/6 
5  2,0 
6  2,4 
7  2,8 
8  3/2 
9  3,6 

955 
956 
957 
958 
959 

98  ooo  005  009  014  019 
046  050  055  059  064 

091  096  loo  105  109 

137  141  146  150  155 

182  186  191  195  200 

023  028  032  037  041 
068  073  078  082  087 
114  118  123  127  132 
159  164  168  173  177 
204  209  214  218  223 

960 

961 
962 
963 
964 

227  232  236  241  24^ 
272  277  281  286  290 
318  322  327  331  336 
363  367  372  376  381 
408  412  417  421  426 

250  254  259  263  268 
295  299  304  308  313 
340  345  349  354  358 
385  390  394  399  403 
430  133  439  444  448 

965 

966 
967 
968 
969 

453  457  462  466  471 
498  502  507  511  516 

543  547  552  556  S^i 
588  592  597  601  605 
632  637  641  646  650 

475  48o  484  489  493 
520  525  529  534  538 

565  57°  574  579  583 
610  614  619  623  628 
655  659  664  668  673 

970 

971 
972 
973 
974 

677  682  686  691  69=; 
722  726  731  735  740 
767  771  776  780  784 
811  816  820  825  829 
856  860  865  869  874 

700  704  709  713  717 
744  749  753  758  762 
789  793  798  802  807 
834  838  843  847  851 
878  883  887  892  896 

976 

977 

978 

979 

900  905  909  914  918 

945  949  954  958  963 
989  994  998  *oo3  *007 
99034  038  043  047  052 
078  083  087  092  096 

923  927  932  936  941 
967  972  976  981  985 
*oi2  *oi6  *o2i  ^025  #029 
056  061  065  069  074 
100  105  109  114  118 

980 

981 
982 

983 
984 

123  127  131  136  140 
167  171  176  180  185 

211   2l6   22O   224   229 
255   260   264   269   273 
300   304   308   313   317 

145  149  154  158  162 

189   193   198   202   207 
233   238   242   247   251 
277   282   286   291   295 

322  326  330  335  339 

985 
986 

987 

988 
989 

344  348  352  357  361 
388  392  396  401  405 
432  436  441  445  449 
476  480  484  489  493 
520  524  528  533  537 

366  370  374  379  383 
410  414  419  423  427 

454  458  463  467  47i 
498  502  506  511  515 
542  546  550  555  559 

990 

•991 
992 
993 
994 

564  568  572  577  581 
607  612  616  621  625 
651  656  660  664  669 
695  699  704  708  712 
739  743  747  752  756 

585  590  594  599  603 
629  634  638  642  647 
673  677  682  686  691 
717  721  726  730  734 

760  765  769  774  778 

995 
996 

997 
998 

999 

782  787  791  79$  800 
826  830  835  839  843 
870  874  878  883  887 
913  917  922  926  930 
957  961  965  970  974 

804  808  813  817  822 
848  852  856  861  865 
891  896  900  904  909 
935  939  944  948  952 
978  983  987  991  996 

1000 

oo  ooo  004  009  013  017 

022   026   030   035   039 

N. 

L.  0   1   2   3   4 

56789 

P.P. 

NOTES   ON   TABLES   I   AND  II. 

The  logarithms  of  numbers  are  in  general  incommensurable. 
In  these  tables  they  are  given  correct  to  five  places  of  decimals. 
If  the  sixth  place  is  5  or  more,  the  next  larger  number  is  used 
in  the  fifth  place.  Thus  log  8102  =  3.908549+;  in  five-place 
tables  this  is  written  3.90855,  the  dash  above  the  5  showing 
that  the  logarithm  is  less  than  given. 

So  log  8133  =  3.910251-;  in  five-place  tables  this  is  written 
3.91025,  the  dot  above  the  5  showing  that  the  logarithm  is  more 
than  given. 

In  the  natural  functions  of  the  angles  (Table  II)  all  numbers 
are  decimals  for  sine  and  cosine  (why  ?),  and  for  tangent  and 
cotangent,  except  where  the  decimal  point  is  used  to  indicate 
that  part  of  the  number  is  integral.  When  no  decimal  point 
is  printed  in  the  tables  it  is  to  be  understood.  When  the 
natural  function  is  a  pure  decimal  the  characteristic  of  the 
logarithm  is  negative.  Accordingly,  in  the  tables  10  is  added, 
and  in  the  result  this  must  be  allowed  for.  Thus 

nat.  sin  44°  20'  =  0.69883,   log  sin  44°  20'  =  1.84437, 
or,  as  printed  in  the  tables,  9.84437,  which  means  9.84437  — 10. 


TABLE   II. 

THE  LOGARITHMIC   AND  NATURAL  SINES,  COSINES, 

TANGENTS,  AND  COTANGENTS  OF  ANGLES 

FROM  0°  TO   90°. 


oc 


f 

Nat.  Sin  Log.   d. 

Nat.CoSLog.|Nat.TanLog.|  c.d. 

Log.CotNat. 

0 

I 

2 

3 

4 

ooooo   — 

029  6.46373 
058  6.76476 
087  6.94085 
116  7.06579 

30103 
17609 
12494 
9691 
7918 
6694 
5800 
5H5 
4576 
4i39 
3779 
3476 
3218 
2997 
2802 
2633 
2483 
2348 
2227 
2119 
202  1 
1930 
1848 

1773 
1704 
1639 
1579 
J524 
1472 
1424 
1379 
1336 
1297 

1259 
1223 
1190 
1158 
1128 

IIOO 

1072 
1046 

IO22 

999 
976 

954 
934 
914 
896 
877 
860 

843 
827 
812 

797 
782 
769 
755 
743 
730 

1  0000  0.00000 
000  0.00000 
000  0.00000 
000  0.00000 
000  O.OOOOO 

ooooo   — 

029  6.46373 
058  6.76470 
087  6.94085 

116  7.06579 

30103 

17609 
12494 
9691 
7918 
6694 
5800 
5"5 
4576 
4139 
3779 
3476 
3219 
2996 
2803 
2633 
2482 
2348 
2228 
2119 

2020 

1931 
1848 

1773 
1704 
I639 

J579 
1524 
1473 
1424 
1379 
1336 
1297 

1259 
1223 
1190 

ii59 
1128 

IIOO 

1072 
1047 

IO22 
998 
976 

955 
934 
915 
895 
878 
860 

843 
828 
812 

797 
782 
769 
756 
742 
730 

3-53627  3437-7 
3.23524  1718.9 
3.05915  1145.9 
2.93421  859.44 

60 

59 
58 

% 

5 

6 

7 
8 

9 
10" 

ii 

12 
13 
14 

15 

16 
17 
18 
19 

00145  7.16270 
175  7.24188 
204  7.30882 
233  7-36682 
262  7.41797 

lOOOO  0.00000 
000  O.OOOOO 
000  O.OOOOO 

ooo  o.ooooo 
ooo  o.ooooo 

00145  7.16270 
175  7.24188 
204  7.30882 
233  7-36682 
262  7.41797 

2.83730  687.55 
2.75812  572.96 
2.69118  491.11 
2.63318  429.72 
2.58203  381.97 

55 

54 
53 
52 
5i 

00291  7.46373 
320  7.5050 

349  7-5429I 
378  7.57767 
407  7.60985 

loooo  o.ooooo 
99999  o.ooooo 
999  o.ooooo 
999  o.ooooo 
999  o.ooooo 

00291  7.46373 
320  7.50512 

349  7-5429I 
378  7.57767 
407  7.60986 

2-53627  343-77 
2.49488  312.52 
2.45709  286.48 
2.42233  264.44 
2.39014  245.55 

50 

49 

4£ 
46 

00436  7.63982 
465  7.66784 
495  7-694I7 
524  7-7I900 
553  7-74248 

99999  o.ooooo 
999  o.ooooo 
999  9.99999 
999  9.99999 
998  9.99999 

00436  7.63982 
465  7.66785 
495  7.69418 
524  7-7I900 
553  7.74248 

2.36018  229.18 
2.33215  214.86 

2.30582  202.22 
2.28100  190.98 
2.25752  180.93 

45 

44 
43 
42 

4i 

20 

21 
22 

23 

24 

25~ 

26 
27 

28 
29 

00582  7.76475 
611  7-78594 
640  7.80615 
669  7.82545 
698  7.8439^5 

99998  9-99999 
998  9-99999 
998  9-99999 
998  9-99999 
998  9-99999 

00582  7.76476 
6«  7-78595 
640  7.80615 
669  7.82546 
698  7.84394 

2.23524  171.89 
2.21405  163.70 
2.19385  156.26 
2.17454  149.47 
2.15606  143.24 

40 

39 
38 
37 

36 

00727  7.86100 
756  7.87870 

785  7-89509 
814  7.91088 
844  7.92612 

99997  9.99999 
997  9.99999 
997  9.99999 
997  9.99999 

00727  7.86167 
756  7.87871 
785  7.89510 
815  7.91089 
844  7.92613 

2.13833  I37-5I 
2.I2I29  132.22 
2.10490  127.32 
2.08911  122.77 
2.07387  118.54 

35 

34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 

24 
23 

22 
21 

20 

19 
18 

17 
16 

30 

31 
32 
33 
34 
35 
36 

? 

39 
40 

4i 

42 

43 

44 

00873  7-940*4 
902  7-95508 
931  7.96887 
960  7.98223 
989  7-99520 

99996  9.99998 
996  9.99998 
996  9.99998 
995  9-99998 
995  9-99998 

00873  7.94086 
902  7-95510 
931  7.96889 
960  7.98225 
989  7.99522 

2.05914  II4-59 
2.04490  110.89 
2.03111  107.43 
2.01775  104.17 

2.00478  ioi.ii 

01018  8.00779 

047  8.02002 
076  8.03192 
105  8.04350 
134  8.05478 

99995  9-99998 
995  9-99998 
994  9-99997 
994  9-99997 
994  9-99997 

01018  8.00781 
047  8.02004 
076  8.03194 
105  8.04353 
135  8.05481 

I.992I9  98.218 
1.97996  95.489 
1.96806  92.908 

I-95647  90.463 
I.945I9  88.144 

01164  8.0057d 
193  8.076gO 
222  8.08696 
251  8.09718 
280  8.I07I7 

99993  9-99997 
993  9-99997 
993  9-99997 
992  9-99997 
992  9.99996 

01164  8.06581 
193  8.07653 

222  8.08700 
251  8.09722 
280  8.10720 

I.934I9  85.940 
1.92347  83.844 
I.9I300  81.847 
1.90278  79.943 
1.89280  78.126 

45 

46 

47 
48 

49 

01309  8.II693 
338  8.12647 

367  8.1358i 
396  8.14495 
425  8.15391 

99991  9.99996 
991  9.99996 
991  9.99996 
990  9.99996 
990  9.99996 

OI3O9  8.11696 
338  8.I265I 
367  8.13585 
396  8.14500 
425  8.15395 

1.88304  76.390 
1-87349  74-729 

1.86415  73-I39 
1.85500  71.615 
1.84605  70.153 

15 

14 

13 

12 
II 

To 

9 
8 

7 
6 

5 

4 
3 

2 

I 

0 

50 

5i 

52 
53 
54 

01454  8.16268 
483  8.17128 

513  8.17971 
542  8.18798 

571  8.19610 

99989  9.9999$ 
989  9.9999$ 
989  9.09995 
988  9.99995 
988  9-99995 

01455  8.16273 
484  8.I7I33 
513  8.17976 
542  8.18804 
571  8.I96I6 

1.83727  68.750 
1.82867  67.402 
1.82024  66.105 
1.81196  64.858 
1.80384  63.657 

55 

56 

P 

§ 

01600  8.20407 
629  8.21189 
658  8.21958 
687  8.22713 
716  8.23456 
745  8.24186 

99987  9.99994 
987  9.99994 
986  9.99994 
986  9.99994 
985  9.99994 
985  9-99993 

OI600  8.20413 
629  8.2II95 
658  8.21964 
687  8.22720 
7l6  8.23462 
746  8.24192 

I-79587  62.499 
1.78805  61.383 
1.78036  60.306 
1.77280  59.266 
1.76538  58.261 
1.75808  57.290 

Nat.  COS  Log.   d. 

Nat.  Sin  Log. 

Nat.CotLog. 

c.d. 

Log.TanNat. 

f 

89° 


r 


f 

Nat.  Sin  Log.    d. 

Nat.  COS  Log.JNat.Tan  Log. 

c.d.  |Log.  Cot  Nat. 

0 

I 

2 

3 
4 

01745    8.24186 
774    8.24903 
803    8.25609 
832    8.26304 
862    8.26988 
01891    8.27661 
920    8.28324 
949    8.28977 
978    8.29621 
02007    8.30255 

717 
706 

695 
684 

673 
663 

653 
644 

634 
624 
616 
608 
599 
590 

583 

575 
568 
560 
553 
547 
539 
533 
526 
520 

Si4 
508 
502 
496 
491 

485 
480 

474 
470 

464 
459 
455 
45° 
445 
441 

436 
433 
427 
424 
419 

416 
411 
408 
404 
400 
396 
393 
390 
386 
382 

379 
376 
373 
369 
367 
363 

99985    9.99993 

984    9-99993 
984    999993 
983    9.99993 
983    9.99992 

01746    8.24192 
775    8.24910 
804    8.25616 
833    8.26312 
862    8.26996 

7II 
706 

696 
684 
673 
663 
654 
643 
634 
625 
617 
607 
599 
59i 
584 

575 
568 

56i 
553 
546 
540 
533 
527 
520 

5H 

5°9 
502 

496 
491 

486 
480 

475 
470 
464 
460 

455 
450 
446 
441 
437 
432 
428 
424 
420 
416 
412 
408 
404 
401 

397 
393 
390 
386 
383 
380 
376 
373 
370 
367 
363 

1.75808    57.290 
1.75090    56.351 
1.74384    55-442 
1.73688    54.561 
1.73004    53.709 

60 

3 

p 

5 

6 

7 
8 

9 

99982    9-99992 
982    9.99992 
981    9.99992 
980   9-99992 
980   9.99991 

01891    8.27669 
920    8.28332 
949    8.28986 
978    8.29629 
02007    8.30263 

I-7233I    52-882 
1.71668       .081 
1.71014   51.303 
I-7«37i    50-549 
I-69737   49-8i6 

55 

54 
53 
52 
5i 

10 

ii 

12 

J3 

14 

IF 

16 

2 

19 

02036    8.30879 
065    8.31495 
094    8.32103 
123    8.32702 
152    8.33292 

99979    9.99991 
979    9.99991 
978    9-99990 
977    9-99990 
977    9-99990 

02036    8.30888 
066    8.31505 
095    8.32112 
124    8.32711 
153    8.33302 

1.69112   49.104 
1.68495   48.412 
1.67888   47.740 
1.67289       .085 
1.66698   46449 

50 

49 
48 

47 
46 

02181    8.33875 

211      8.34450 
240      8.35018 
269      8.35578 
298      8.36I3I 
02327      8.36678 
356      8.37217 
385      8.37750 
414     8.38276 

443    8.38796 

99976    9.99990 
976    9.99989 
975    9-99989 
974    9.99989 
974    9.99989 

02182    8.33886 

211      8.34461 
240      8.35029 

298      8.36143 

1.66114   45.829 
1.65539       .226 
1.64971    44.639 
1.64410       .066 
1-63857   43-5o8 

45 

44 
43 
42 
4i 

20 

21 

22 

23 
24 

99973    9-99988 
972    9.99988 
972    9.99988 
971    9.99987 
970    9.99987 

02328      8.36689 

357    8.37229 
386    8.37762 
415    8.38289 
444    8.38809 

1.63311    42.964 
1.62771        .433 
1.62238    41.916 
1.61711        .411 
1.61191    40.917 

40 

39 
38 
37 
36 

25 

26 
27 
28 
29 

02472    8.39310 
501    8.39818 
530    840320 
560    8.40816 
589    841307 

99969    9.99987 
969    9.99986 
968    9.99986 
967    9.99986 
966    9.99985 

02473    8.39323 
502    8.39832 
'    531    840334 
560    8.40830 
589    841321 

1.60677    40-436 
1.  60168    39.965 
1.59666        .506 
1.59170        .057 
1.58679    38.618 

35 

34 
33 
32 
3i 

30 

31 
32 
33 
34 

02618    841792 
647    8.42272 
676    842746 
705    843216 
734    843680 

99966    9.99985 
965    9.99985 
964    9.99984 

963    9-99984 
963    9.99984 

02619    841807 
648    8.42287 
677    8.42762 
706    843232 
735    843696 

I-58I93    38.188 
i-577!3   37-769 
1-57238       -358 
1.56768   36.956 
1.56304       .563 

30 

29 
28 

27 
26 

35 

36 

s 

39 
40 

41 
42 

43 
44 

02763    8.44139 
792    8.44594 
821    8.45044 
850    845489 
879    845930 

99962    9.99983 
961    9.99983 
960    9.99983 
959    9-99982 
959    9-99932 

02764    8.44156 
793    8.44611 
822    845061 
851    845507 
88  1    845948 

1.55844  36.178 
^55389   35-8oi 
1.54939       .431 
1.54493       .070 
1.54052   34.715 

25 

24 
23 

22 
21 

02908    846366 
938    846799 
967    847226 
996    847650 
03025    8.48069 

99958    9.99982 
957    9-9993I 
956    9-9998I 
955    9-9998I 
954    9-99980 

02910    846385 
939    846817 
968    8.47245 
997    847669 
03026    8.48089 

1-53615   34.368 
1.53183       .027 
I-52755   33-694 
1-52331       -366 
1.51911       .045 

20 

19 

18 

17 
16 

45 

46 
47 
48 

49 

03054    848485 
083    8.48896 

112      8.49304 
141      849708 
170     8.5OIO8 

99953    9-99980 
952    9-99979 
952   9-99979 
95i    9-99979 
950   9.99978 

03055    848505 
084    848917 
114    849325 
143    8.49729 
172    8.50130 
03201    8.50527 
230    8.50920 
259    8.51310 
288    8.51696 
317    8.52079 

I-5I495   32-730 
1.51083       .421 
1.50675       .118 
1.50271    31.821 
149870       .528 

15 

14 
13 

12 
II 

50 

5i 
52 
53 
54 

03199    8.50504 
228    8.50897 
257    8.51287 
286    8.51673 
316    8.52055 

99949    9-99978 
948    9.99977 
947    9-99977 
946    9.99977 
945    9-99976 

M9473   3J-242 
1.49080    30.960 
148690       .683 
148304       .412 
147921       .145 

10 

I 

I 

"5" 

4 
3 

2 

I 

0 

55 

56 
57 

58 

ft 

03345    8.52434 
374    8.52810 
403    8.53183 
432    8.53552 
461    8.53919 
490    8.54282 

99944   9-99976 
943    9-99975 
942    9-99975 
941    9.99974 
940   9-99974 
939    9-99974 

03346    8.52459 

405    8.53208 
434    8.53578 
463    8.53945 
492    8.54308 

1.47541    29.882 
147165       .624 
146792       .371 

1.46422           .122 
1.46055     28.877 
145692           .636 

Nat.  COS  Log.    d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d. 

Log.  Tan  Nat. 

f 

88C 


2° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log. 

Nat.Tan  Log.  c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

03490  8.54282 
519  8.54642 
548  8.54999 

577  8-55354 
606  8.55705 

360 
357 
355 
35i 
349 
346 
343 
34i 
337 
336 
332 
330 
328 
325 
323 
320 
3i8 
316 
313 
3" 
309 
307 
305 
302 

301 
298 
296 
294 
293 
290 
288 
287 
284 
283 
281 
279 
277 
276 
274 
272 
270 
269 
267 
266 
263 
263 
260 
259 
258 

256 
254 
253 
252 
250 

249 
247 
246 
244 

243 
242 

99939  999974 
938  9.99973 

937  9-99973 
936  9-99972 
935  9-99972 

03492  8.54308 
521  8.54669 
550  8.55027 
579  8.55382 
609  8.55734 

361 

358 
355 
352 
349 
346 
344 
34i 
338 
336 
333 
330 
328 
326 

323 
321 
3J9 
316 
3H 

3" 

310 

307 
305 
303 
301 
299 
297 
295 
292 

291 

289 
287 
285 
284 
281 
280 
278 
276 
274 

273 
271 
269 
268 
266 
264 
263 
261 
260 
258 
257 
255 
254 
252 

251 
249 
248 
246 
245 
244 
243 

145692  28.636 
I-4533I   -399 
1.44973   -l66 
1.44618  27.937 
1.44266   .712 

60 

59 
58 
57 
56 

5 

6 

I 

9 

°3635  8-56054 
664  8.56400 
693  8.56743 
723  8.57084 
752  8.57421 

99934  9-99971 
933  9-99971 
932  9-99970 
93i  9-99970 
930  9.99969 

03638  8.56083 
667  8.56429 
696  8.56773 
725  8.57114 
754  8.57452 

03783  8.57788 
812  8.58121 
842  8.58451 
871  8.58779 
900  8.59105 

143917  27.490 
143571   .271 
1.43227   .057 
1.42886  26.845 
1.42548   .637 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

03781  8.57757 
8  10  8.58089 
839  8.58419 
868  8.58747 
897  8.59072 

99929  9-99969 
927  9.99968 
926  9.99968 
925  9-99967 
924  9-99967 

1.42212  26.432 
1.41879   .230 
1.41549   .031 
141221  25.835 
1.40895   .642 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

03926  8.5939$ 
955  8.59715 
984  8.60033 
04013  8.60349 
042  8.60662 

99923  9.99967 
922  9.99966 

92!  9.99966 
919  9.99965 
9I8  9.99964 

03929  8.59428 
958  8.59749 
987  8.60068 
04016  8.60384 
046  8.60698 

140572  25.452 
1.40251   .264 
I«39932   -080 
1.39616  24.898 
1.39302   .719 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 

20 

21 
22 

23 
24 

04071  8.60973 
loo  8.61282 
129  8.61589 
159  8.61894 
188  8.62196 

99917  9.99964 
9l6  9.99963 
915  9.99963 
913  9.99962 
912  9.99962 

04075  8.61009 
104  8.61319 
133  8.61626 
162  8.61931 
191  8.62234 

1.38991  24.542 
1.38681   .368 
1.38374   .196 
1.38069   .026 
1.37766  23.859 

25 

26 
27 
28 
29 

"80 

3i 
32 
33 

34 

04217  8.62497 
246  8.62795 
275  8.63091 
304  8.63385 
333  8.63678 

999II  9.99961 
910  9.99961 
909  9.99960 
907  9.99960 
906  9-99959 

04220  8.62535 
250  8.62834 
279  8.63131 
308  8.63426 
337  8.63718 

1.37465  23.695 
1.37166   .532 
1.36869   .372 
1.36574   .214 
1.36282   .058 

04362  8.63968 
391  8.64256 
420  8.64543 
449  8.64827 
478  8.65110 

99905  9-99959 
904  9.99958 
902  9.99958 
901  9-99957 
900  9.99956 

04366  8.64009 
395  8.64298 
424  8.64585 
454  8.64870 
483  8.65154 

1.3599!  22.904 
1.35702   .752 
1.35415   .602 
i-35I30   -454 
1.34846   .308 

30 

29 
28 
27 
26 

35 

36 
37 
38 
39 

04507  8.65391 

565  8.65947 
594  8.66223 
623  8.66497 

99898  9-99956 

897  9-99955 
896  9-99955 
894  9.99954 
893  9-99954 

04512  8.65435 
541  8.65715 
570  8.65993 
599  8.66269 
628  8.66543 

1-34565  22.164 

1.34285     .022 
I.340O7   2I.88I 

I-3373I   -743 
1-33457   •6o6 

25 

24 
23 

22 
21 

40 

4i 
42 
43 
44 
45 
46 

47 
48 

49 
50 

5i 
52 
53 
54 

04653  8.66769 
682  8.67039 
711  8.67308 
740  8.67575 
769  8.6^841 

99892  9-99953 
890  9-99952 
889  9-99952 
888  9-99951 
886  9.99951 

04658  8.66816 
687  8.67087 
716  8.67356 
745  8.67624 
774  8.67890 

1.33184  21.470 
1.32913   .337 
1.32644   .205 
1.32376   .075 
1.32110  20.946 

20 

19 
18 

17 
16 

04798  8.68104 
827  8.68367 

885  8.68886 
914  8.69144 

99885  9.99950 
883  9-99949 

879  9.99948 

04803  8.68154 
833  8.68417 
862  8.68678 
891  8.68938 
920  8.69196 

1.31846  20.819 

i-s^s  -693 
1.31322  .569 

1.31062   .446 

1.30804  .325 

15 

14 
13 

12 
II 

04943  8.69400 
972  8.69654 
05001  8.69907 
030  8.70159 
059  8.70409 

99878  9-99947 
876  9.99946 
875  9-99946 
873  9-99945 
872  9.99944 

04949  8.69453 
978  8.69708 
05007  8.69962 
°37  8.70214 
066  8.70465 

1.30547  20.206 
1.30292   .087 
1.30038  19.970 
1.29786   .855 
1-29535  .740 

10 

9 

8 

I 

55 

56 

11 
11 

05088  8.70658 
117  8.70905 
146  8.71151 
J75  8.71395 
205  8.71638 
234  8.71880 

99870  9-99944 
869  9-99943 
867  9.99942 
866  9.99942 
864  9.99941 
863  9-99940 

05095  8.70714 
124  8.70962 
153  8.71208 
182  8.71453 

212  8.71697 
241  8.71940 

1.29286  19.627 
1.29038   .516 
1.28792   .405 
1.28547   .296 
1.28303  .188 
1.28060   .081 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d.  1  Log.  Tan  Nat. 

t 

87C 


3° 


t 

Nat.  Sin  Log. 

d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

05234  8.71880 
263  8.72120 

292  8.72359 
321  8.72597 
350  8.72834 

240 

239 
238 
237 

99863  9.99940 
861  9-99940 
860  9.99939 
858  9.99938 
857  9-99938 

05341  8.71940 
270  8.72181 
299  8.72420 

328  8.72659 

357  8.72896 

241 
239 
239 
237 
2o6 

1.28060  19.081 
1.27819  18.976 
1.27580   .871 
1.27341   .768 
1.27104   .666 

60 

59 
58 

i 

5 

6 

7 
8 

9 

05379  8.73069 

408  8.73303 
437  8.73535 
466  8.73767 
495  8.73997 

235 
234 
232 
232 
230 

99855  9-99937 
854  9-99936 
852  9.99936 
851  9.99935 
849  9-99934 

05387  8.73132 
416  8.73366 
445  8.73600 
474  8.73832 
503  8.74063 

430 

234 
234 
232 
231 

1.26868  18.564 
1.26634   .464 
1.26400   .366 
1.26168   .268 
1-25937   -171 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 

14 

05524  8.74226 
553  8.74454 
582  8.74680 
611  8.74906 
640  8.75130 

229 
228 
226 
226 
224 

99847  9.99934 
846  9-99933 
844  9-99932 
842  9.99932 
841  9-99931 

05533  8.74292 
562  8.74521 
591  8.74748 
620  8.74974 
649  8.75199 

229 
227 
226 
225 

1.25708  18.075 
1.25479  17.980 
1.25252   .886 
1.25026   .793 
1.24801   .702 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

05669  8.75353 
698  8.75575 
727  8.7579$ 
756  8.76015 
785  8.76234 

223 

222 
22O 
22O 
219 

99839  9-99930 
838  9.99929 
836  9-99929 
834  9.99928 
833  9-9»27 

05678  8.75423 
708  8.75645 
737  8.75867 
766  8.76087 
795  8.76306 

222 
222 
22O 
219 

1.24577  17.611 

1.24133   .431 

I.239I3   -343 
1.23694   .256 

45 

44 
43 
42 
4i 

20 

21 
22 

23 
24 

05814  8.76451 
844  8.76667 
873  8.76883 
902  8.77097 
931  8.77310 

217 

216 
216 

2I4 
2I3 

99831  9.99926 
829  9.99926 
827  9.99925 
826  9-99924 
824  9-99923 

05824  8.76525 
854  8.76742 
883  8.76958 
912  8.77173 
941  8.77387 

219 
217 

216 

215 
214 

1.23475  17.169 
1.23258   .084 
1.23042  16.999 
1.22827   .915 
1.22613   .832 

40 

P 

i 

25 

26 
27 
28 
29 

05960  8^77522 
989  8.77733 
06018  8.77943 
047  8.78152 
076  8.78360 

211 
210 
209 
208 
2O8 

99822  9.99923 

82!  9.99922 

819  9.99921 
817  9.99920 
815  9.99920 

0597°  8.77600 
999  8.77811 
06029  8.78022 
058  8.78232 
087  8.78441 

^i5 

211 
211 
210 
209 
2O8 

1.22400  16.750 
1.22189   .668 
1.21978   .587 
1.21768    .507 
1.21559   .428 

35 

34 
33 
32 
3i 

30 

31 
32 
33 

34 

06105  8.78568 
134  8.78774 
163  8.78979 
192  8.79183 

221  8.79386 

206 
205 
204 
203 

99813  9-99919 

8  12  9.99918 
8  10  9.99917 

808  9-99917 

806  9.99916 

06116  8.78649 
145  8.78855 
175  8.79061 
204  8.79266 
233  8.79470 

206 
206 
205 
204 

1.21351  16.350 
1.21145    -272 
1.20939   -J95 
1.20734   .119 
1.20530   .043 

30 

29 

28 

Z 

35 

36 

P 

39 

06250  8.79588 
279  8.79789 
308  8.79990 

337  8.80189 
366  8.80388 

201 
201 
I99 
I99 

99804  9.99915 
803  9.99914 

801  9.99913 
799  9.99913 
797  9-99912 

06262  8.79673 
291  8.79875 
321  8.80076 
350  8.80277 
379  8.80476 

203 
202 
201 
201 
199 

108 

1.20327  15.969 
1.20125   .895 
1.19924   .821 
1.19723   .748 
1.19524   .676 

25 

24 
23 

22 
21 

40 

4i 
42 

43 
44 

06395  8.80585 
424  8.80782 
453  8.80978 
482  8.81173 
511  8.81367 

197 
197 
I96 

195 
194 

99795  9-999" 
793  9-999Jo 
792  9.99909 
790  9-99909 
788  9.99908 

06408  8.80674 
438  8.80872 
467  8.81068 
496  8.81264 
525  8.81459 

198 
196 
196 
J95 

1.19326  15.605 
1.19128   .534 
1.18932   .464 
1.18736   .394 
1.18541   .325 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

06540  8.81560 
569  8.81752 
598  8.81944 
627  8.82134 
656  8.82324 

J93 
192 
192 
190 
190 
1  80 

99786  9.99907 
784  9.99906 
782  9.99905 
780  9.99904 
778  9.99904 

06554  8.81653 
584  8.81846 
613  8.82038 
642  8.82230 
671  8.82420 

194 

193 
192 
192 
190 

1.18347  15.257 
1.18154   .189 

I.I7962     .122 
I.I7770     .056 
I.I7580   14.990 

15 

H 
13 

12 
II 

50 

5i 
52 

53 
54 

06685  8.82513 
714  8.82701 
743  8.82888 
773  8.83075 
802  8.83261 

±oy 
188 
187 
187 
186 

TQ~ 

99776  9.99903 
774  9-99902 
772  9.99901 
770  9.99900 
768  9.99899 

06700  8.82610 
730  8.82799 
759  8.82987 
788  8.83175 
817  8.83361 

189 
188 
188 
186 

To/^ 

I.I7390   14.924 
I.I720I     .860 
I.I70I3     .795 
I.I6825     .732 
I.I6639     -669 

10 

9 

8 

i 

55 

56 

3 

8 

06831  8.83446 
860  8.83630 
889  8.83813 
918  8.83996 
947  8.84177 
976  8.84358 

M  HI  HI  H  HI  h 
OO  OO  OO  00  00  C 
M  MOOOO-P^  0 

99766  9.99898 
764  9.99898 
762  9.99897 
760  9.99896 
758  9.99895 
756  9-99894 

06847  8.83547 
876  8.83732 
905  8.83916 
934  8.84100 
963  8.84282 
993  8.84464 

185 
184 
184 
182 
182 

I.I6453   14.606 
I.I6268     .544 
I.I6084     .482 
I.I5900     .421 
I.I57I8     .361 
LI5536     .301 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log. 

d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d. 

|Log.  Tan  Nat. 

f 

86( 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log 

Nat.Tan  Log.  c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 
5 
6 

9 

06976  8.84358 
07005  8.84539 
034  8.84718 
063  8.84897 
092  8.85075 

181 
179 
179 
178 
177 
177 
176 
175 
175 
173 
173 
173 
171 
171 
171 
169 
169 
169 
167 
168 
166 
166 
165 
164 

164 

I63 

163 
162 
162 
160 
161 
159 
159 
159 
158 
J57 
157 
156 
155 
155 
155 
154 
153 
153 
152 
152 
151 
151 
J5o 
150 
149 
149 
148 
147 
147 
147 
146 
146 
145 
145 

99756  9-99894 
754  9-99893 
752  9.99892 
750  9.99891 
748  9.99891 

06993  8.84464 

07022  8.84646 
051  8.84826 
080  8.85006 
no  8.85185 

182 
180 
180 
179 
178 
177 
177 
176 
176 
174 
174 

174 
172 
172 
171 
171 
170 
169 
169 
168 
167 
167 
166 
165 
165 

^5 
163 
163 
163 
161 
162 
160 
160 
160 
159 
158 
158 
157 
157 
156 
155 
155 
155 
153 
154 
153 
152 
IS2 
151 
I5i 
150 
15° 
149 
148 
149 
'147 
147 
147 
146 
146 

1.15536  14.301 
1.15354  .241 
1.15174  .182 
1.14994  .124 
1.14815  .065 

60 

59 
58 

11 

07121  8.85252 
150  8.85429 
*79  8.85605 
208  8.85780 
237  8.85955 

99746  9.99890 
744  9.99889 
742  9.99888 
740  9.99887 
738  9.99886 

07139  8.85363 
1  68  8.85540 
197  8.85717 

1.14637  14.008 
1.14460  13.951 
1.14283  .894 
1.14107  .838 
1.13931  .782 

55 

54 
53 
52 
5i 

10 

ii 

12 

*3 
14 

07266  8.86128 
295  8.86301 
324  8.86474 
353  8.86645 
382  8.86816 

99736  9.99885 
734  9.99884 
731  9.99883 
729  9.99882 
727  9.99881 

07285  8.86243 
314  8.86417 
344  8.86591 
373  8.86763 
402  8.86935 

I-I3757  J3-727 
1.13583  .672 
1.13409  .617 

I.I3237  -563 
1.13065  .510 

50 

49 
48 
47 
46 

15 

16 

17 

18 

19 

07411  8.86987 
440  8.87156 
469  8.87325 
498  8.87494 
527  8.87661 

99725  9-99880 
723  9.99879 
721  9.99879 
719  9.99878 
716  9.99877 

07431  8.87106 
461  8.87277 
490  8.87447 
519  8.87616 
548  8.87785 

1.12894  13-457 
1.12723  .404 
1.12553  .352 
1.12384  .300 
1.12215  -248 

45 

44 
43 
42 
4i 

20 

21 
22 

23 
24 

07556  8.87829 

614  8.88161 
643  8.88326 
672  8.88490 

99714  9.99876 
712  9-99875 
710  9.99874 
708  9.99873 
705  9.99872 

07578  8.87953 
607  8.88120 
636  8.88287 
665  8.88453 
695  8.88618 

1.12047  I3.I97 
1.11880  .146 
1.11713  .096 
1.11547  .046 
1.11382  12.996 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 

22 
21 

25 

26 
27 

28 
29 

07701  8.88654 
730  8.88817 
759  8.88980 
788  8.89142 
817  8.89304 

99703  9.99871 
701  9.99870 
699  9.99869 
696  9.99868 
694  9.99867 

07724  8.88783 
753  8.88948 
782  8.89111 
812  8.89274 
841  8.89437 

1.11217  12.947 
1.11052  .898 
1.10889  -85° 
1.10726  .801 
1.10563  .754 

30 

3i 
32 
33 
34 

07846  8.89464 
875  8.89625 
904  8.89784 
933  8.89943 
962  8.90102 

99692  9.99866 
689  9.99865 
687  9.99864 
685  9-99863 
683  9.99862 

07870  8.89598 
899  8.89760 
929  8.89920 
958  8.90080 
987  8.90240 

1.10402  12.706 
1.10240  .659 
1.10080  .612 
1.09920  .566 
1.09760  .520 

35 

36 

H 

39 

07991  8.90260 
08020  8.90417 
049  8.90574 
078  8.90730 
107  8.90885 

99680  9.99861 
678  9.99860 
676  9.99859 
673  9.99858 
671  9.99857 

08017  8.90399 
046  8.90557 
075  8.90715 
104  8.90872 
134  8.91029 

1.09601  12.474 
1.09443  .429 
1.09285  .384 
1.09128  .339 
1.08971  .295 

40 

41 
42 
43 
44 

08136  8.91040 
165  8.91195 
194  8.91349 
223  8.91502 
252  8.91655 

99668  9.99856 
666  9.99855 
664  9-99854 
661  9.99853 
659  9.99852 

08163  8.91185 
192  8.91340 

221   8.91495 
251   8.91650 
280  8.91803 

1.08815  12.251 
1.08660  .207 
1.08505  .163 

1.08350  .120 
I.08I97  .077 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

08281  8.91807 
310  8.91959 
339  8.92110 
368  8.92261 
397  8.92411 

99657  9-99851 
654  9-99850 
652  9.99848 

649  9-99847 
647  9.99846 

08309  8.91957 

339  8.92110 
368  8.92262 
397  8.92414 
427  8.92565 

1.08043  12.035 
1.07890  11.992 
1.07738  .950 
1.07586  .909 
1.07435  .867 

15 

14 
13 

12 
II 

50 

5i 

52 
53 
54 

08426  8.92561 
455  8.92710 
484  8.92859 
513  8.93007 
542  8.93154 

99644  9-99845 
642  9-99844 
639  9.99843 
637  9-99842 
635  9.9984I 

08456  8.92716 
485  8.92866 
514  8.93016 
544  8.93165 
573  8.93313 

1.07284  11.826 
I.07I34  .785 
1.06984  .745 

1.06687  .664 

10 

9 
8 

7 
6 

~5 

4 
3 

2 

0 

55 

56 

3 

$ 

08571  8.93301 
600  8.93448 
629  8.93594 
658  8.93740 
687  8.93885 
716  8.94030 

99632  9.99840 
630  9.99839 
627  9.99838 
625  9.99837 
622  9.99836 
619  9.99834 

08602  8.93462 
632  8.93609 
66  1  8.93756 
690  8.93903 
720  8.94049 
749  8.94195 

1.06538  11.625 
I.0639I  .585 
1.06244  .546 
1.06097  .507 
I.0595I  .468 
1.05805  .430 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d.  Log.  Tan  Nat. 

r 

85C 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 
5 

6 

7 
8 

9 

08716  8.94030 
745  8.94174 
774  8.94317 
803  8.94461 
831  8.94603 

144 

143 
144 
142 

143 
141 
142 
141 
140 
140 
139 
139 
139 
138 
138 
i37 
137 
136 
136 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 

I31 
131 
131 
130 
130 
129 
129 
129 
128 
128 
128 
127 
127 
126 
I26 
126 
125 
I25 
124 

125 
123 
124 
123 
123 

122 
122 
122 
121 
121 
121 
1  2O 

99619  9.99834 
617  9.99833 
614  9.99832 
612  9.99831 
609  9.99830 

08749  8.94195 
778  8.94340 
807  8.94485 
837  8.94630 

866  8.94773 

J45 
145 
145 
M3 
144 

143 
142 
142 
142 
141 
140 
141 

139 
140 

138 
139 
138 
137 
138 
136 
137 
135 
136 

135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
130 
130 
130 
129 
128 
129 
128 
127 
128 
127 
126 
126 
126 
126 
125 

I25 
124 
124 
124 
123 
123 
123 

122 
122 
122 

1.05805  11.430 
1.05660   .392 
I.055I5   -354 
1.05370   .316 
1.05227   .279 

60 

P 
1 

08860  8.94746 
889  8.94887 
918  8.95029 
947  8.95170 
976  8.95310 

99607  9.99829 
604  9.99828 
602  9.99827 
599  9.99825 
596  9.99824 

08895  8.94917 

925  8.95060 
954  8.95202 
983  8.95344 
09013  8.95486 

1.05083  11.242 
1.04940   .205 
1.04798   .168 
1.04656   .132 
1.04514   .095 

55 

54 
53 
52 
5i 
50 

3 

1 

45 

44 
43 
42 

41 

10 

II 

12 
13 
14 

09005  8.95450 
°34  8.95589 
063  8.95728 
092  8.95867 

121  8.96005 

99594  9-99823 
591  9.99822 
588  9.99821 
586  9.99820 
583  9.99819 

09042  8.95627 
071  8.95767 
101  8.95908 
130  8.96047 
159  8.96187 

1.04373  "-059 
1.04233   .024 
1.04092  10.988 

1-03953   -953 
1.03813   .918 

15 

16 

2 

19 

09150  8.96143 
179  8.96280 
208  8.96417 
237  8.96553 
266  8.96689 

99580  9.99817 
578  9.99816 

575  9-998I5 
572  9.99814 
570  9.99813 

09189  8.96325 
218  8.96464 
247  8.96602 
277  8.96739 
306  8.96877 

1.03675  10.883 
1.03536   -848 
1.03398   .814 
1.03261   .780 
1.03123   .746 

20 

21 
22 

23 
24 

09295  8.96825. 
324  8.96960 

353  8.97095 
382  8.97229 
411  8.97363 

99567  9-998I2 
564  9.99810 
562  9.99809 
559  9.99808 
556  9-99807 

09335  8.97013 
365  8.97150 
394  8.97285 
423  8.97421 
453  8.97556 

1.02987  10.712 
1.02850   .678 
1.02715   .645 
1.02579   .612 
1.02444   .579 

40 

P 

% 

25 

26 

27 
28 
29 

09440  8.97496 
469  8.97629 
498  8.97762 
527  8.97894 
556  8.98026 

99553  9-998o6 
551  9.99804 
548  9.99803 
545  9.99802 
542  9.99801 

09482  8.97691 
511  8.97825 
541  8.97959 
570  8.98092 
600  8.98225 

1.02309  10.546 
1.02175   .514 
1.02041   .481 
1.01908   .449 
1.01775   .417 

35 

34 
33 
32 
•3i 

30 

3i 
32 
33 
34 
35- 
36 

P 

39 

09585  8.98157 
614  8.98288 
642  8.98419 
671  8.98549 
700  8.98679 

99540  9.99800 
537  9-99798 
534  9-99797 
53i  9-99796 
528  9-99795 

09629  8.98358 
658  8.98490 
688  8.98622 
717  8.98753 
746  8.98884 

1.01642  10.385 
1.01510   .354 
1.01378   .322 
1.01247   .291 
1.01116   .260 

30 

29 
28 
27 
26 

09729  8.98808 
758  8.98937 
707  8.99066 
816  8.99194 
845  8.99322 

99526  9.99793 
523  9.99792 
520  9.99791 
517  9.99790 
514  9.99788 

09776  8.99015 
805  8.99145 
834  8.99275 
864  8:99405 
893  8.99534 

1.00985  10.229 
1.00855   •I99 
1.00725   .168 
1.00595   -138 
1.00466   .108 

25 

24 
23 

22 
21 

40 

41 

42 

43 
44 
45 

46 

47 
48 

49 

09874  8.99450 
9°3  8.99577 
932  8.99704 
961  8  99830 
990  8.99956 

99511  9.99787 
508  9-99786 
506  9-99785 
503  9.99783 
500  9.99782 

09923  8.99662 
952  8.99791 
981  8.99919 
loo  1  1  9.00046 
040  9.00174 

1.00338  10.078 
1.00209   .048 
1.00081   .019 

0-99954  9-9893 
0.99826    601 

20 

19 
18 

17 
16 

10019  9.00082 
048  9.00207 
077  9-00332 
106  9.00456 
135  9.00581 

99497  9.99781 
494  9-99780 
491  9-99778 
488  9-99777 
485  9-99776 

10069  9.00301 
099  9.00427 
128  9.00553 

0.99699  9.9310 
0-99573   021 
0-99447  9.8734 
0.99321    448 
0.99195    164 

15 

14 
13 

12 
II 

TO 

9 
8 

| 

50 

5i 
52 
53 
54 

10164  9.00704 
192  9.00828 

221  9.00951 
250  9.01074 
279  9.0II96 

99482  9.99775 
479  9-99773 
476  9.99772 

473  9-99771 
470  9.99769 

10216  9.00930 
246  9.01055 
275  9.01179 
305  9.01303 
334  9.01427 

0.99070  9.7882 
0.98945    601 
0.98821    322 
0.98697    044 
0.98573  9-6768 

55 

56 
57 
58 

ft 

10308  9.0I3I8 

337  9-OI440 
366  9.01561 
395  9.01682 
424  9.01803 
453  9-01923 

99467  9.99768 
464  9.99767 
461  9.99765 
458  9-99764 
455  999763 
452  9-9976i 

10363  9.01550 
393  9.01673 
422  9.01796 
452  9.01918 
481  9.02040 
510  9.02162 

0.98450  9.6493 

0.98327     220 
0.98204  9.5949 
0.98082     679 
0.97960     411 
0.97838     144 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d.  Log.  Tan  Nat. 

t 

84C 


6C 


f 

Nat.  Sin  Log. 

d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

10453  9-01923 
482  9.02043 
511  9.02163 
540  9.02283 
569  9.02402 

120 
120 
120 
119 
TtR 

99452  9.99761 
449  9.99760 
446  9-99759 
443  9-99757 
440  9.99756 

10510  9.02162 

540  9.02283 

569  9.02404 

121 
121 
121 
120 

0.97838  9.5144 
0.97717  9.4878 
0.97596   614 
0.97475   352 
0-97355   090 

60 
P 
1 

5 

6 

7 
8 

9 

10597  9-02520 
626  9.02639 

655  9.02757 
684  9.02874 
713  9.02992 

119 
118 

117 

118 

99437  9-99755 
434  9-99753 
431  9.99752 
428  9-99751 
424  9.99749 

10657  9.02766 
687  9.02885 
716  9.03005 

746  9.03124 

775  9.03242 

119 
120 
119 

118 

0.97234  9.3831 
o-97"5   572 
0-96995   3*5 
0.96876   060 
0.96758  9.2806 

55 

54 
53 
52 

10 

ii 

12 
13 

14 

10742  9.03109 
771  9.03226 
800  9.03342 
829  9.03458 
858  9-03574 

117 
116 
116 
116 
116 

99421  9.99748 
418  9.99747 

415  9-99745 
412  9.99744 
409  9.99742 

10805  9.03361 
834  9-03479 
863  9.03597 
893  9.03714 
922  9.03832 

119 
118 
118 
117 
118 
116 

0.96639  9.2553 
0.96521   302 
0.96403   052 
0.96286  9.1803 
0.96168    555 

50 

49 
48 
47 
46 

15 

16 

17 
18 

19 

10887  9.03690 
916  9.03805 
945  9.03920 
973  9.04034 

1  1002  9.04149 

"5 
"5 
114 

99406  9.99741 
402  9.99740 
399  9-99738 
396  9.99737 
393  9-99736 

10952  9.03948 
981  9.04065 
lion  9.04181 
040  9.04297 
070  9.04413 

117 
116 
116 
116 

0.96052  9.1309 
095935    065 
0.95819  9.0821 
0.95703    579 
0.95587    338 

45 
44 
43 
42 
41 

20 

21 
22 

23 
24 

II03I  9.04262 

060  9.04376 

089  9.04490 

118  9.04603 
147  9.04715 

114 

114 

112 

99390  9.99734 
386  9-99733 
383  9-99731 
380  9.99730 
377  9.99728 

11099  9.04528 
128  9.04643 
158  9.04758 
187  9.04873 
217  9.04987 

•<  H  IH  M  H  I- 

?  £cnCnCn  O 

0.95472  9.0098 
0-95357  8.9860 
0.95242   623 
0.95127   387 
0.95013   152 

40 

39 
38 
37 
36 

25 

26 

27 
28 
29 

11176  9.04828 
205  9.04940 
234  9-05052 
263  9.05164 
291  9.05275 

112 
112 
112 
III 

99374  9.99727 
370  9.99726 
367  9.99724 
364  9.99723 
360  9.99721 

11246  9.05101 
276  9.05214 
305  9.05328 
335  9-0544I 
364  9-05553 

XiiJ. 

114 

"3 

112 

0.94899  8.8919 
0.94786    686 
0.94672    455 
0.94559    225 
0.94447  8.7996 

35 

34 
33 

32 

30 

32 
33 
34 

11320  9.05386 
349  9.05497 
378  9-05607 
407  9.05717 
436  9.05827 

III 
IIO 
IIO 
IIO 

99357  9-99720 
354  9-997I8 
351  9.997I7 
347  9.99716 
344  9.99714 

11394  9.05666 
423  9.05778 
452  9.05890 
482  9.06002 
511  9.06113 

112 
112 
112 
III 

0.94334  8.7769 
0.94222    542 
0.94110    317 
0.93998    093 
0.93887  8.6870 

30 

29 
28 
27 
26 

35 

36 

P 

39 

11465  9.05937 
494  9.06046 
523  9.06155 
552  9.06264 
580  9.06372 

109 
109 
109 

108 

99341  9.99713 
337  9-997II 
334  9.99710 
331  9.99708 
327  9.99707 

11541  9.06224 
570  9.0633^ 
600  9.06445 
629  9.06556 

III 
IIO 
III 
IIO 

0.93776  8.6648 
0.93665    427 
0-93555   208 
0.93444  8.5989 
0.93334   772 

25 

24 
23 

22 
21 

40 

42 
43 

44 

11609  9.06481 
638  9.06589 
667  9.06696 
696  9.06804 
725  9.06911 

109 
108 
107 
108 
107 

99324  9.99705 
320  9.99704 
317  9.99702 
314  9.99701 
310  9.99699 

11688  9.06775 
718  9.06885 
747  9.06994 
777  9.07103 
806  9.07211 

IO9 
IIO 
109 
109 

108 

0.93225  8.5555 
0.93115   340 
0.93006   126 
0.92897  8.4913 
0.92789   701 

20 

19 
18 

45 

46 

47 
48 

49 

11754  9.07018 
783  9.07124 
812  9.07231 
840  9.07337 
869  9.07442 

107 
106 
107 
106 

99307  9-99698 
303  9.99696 
300  9.99695 
297  9.99693 
293  9.99692 

11836  9.07320 
865  9.07428 
895  9-07536 
924  9.07643 

954  9-07751 

109 
108 
108 
107 
108 

0.92680  8.4490 
0.92572   280 
0.92464   071 
0.92357  8.3863 
0.92249   656 

15 

14 
13 

12 
II 

50 

51 
52 
53 
54 

11898  9.07548 
927  9.07653 
956  9.07758 
985  9.07863 
12014  9.07968 

105 
i°5 

99290  9.99690 
286  9.99689 
283  9.99687 
279  9.99686 
276  9.99684 

11983  9.07858 
12013  9.07964 
042  9.08071 
072  9.08177 
101  9.08283 

107 
106 

106 

0.92142  8.3450 
0.92036   245 
0.91929   041 
0.91823  8.2838 
0.91717   636 

10 

7 
6 

55 

56 

1 

12043  9.08072 
071  9.08176 
loo  9.08280 
129  9.08383 
158  9.08486 
187  9.08589 

104 
104 
103 
103 
103 

99272  9.99683 
269  9.99681 
265  9.99680 
262  9.99678 
258  9.99677 
255  9-99675 

12131  9.08389 
160  9.08495 
190  9.08600 
219  9.08705 
249  9.08810 
278  9.08914 

106 
105 
104 

0.91611  8.2434 
0-91505   234 
0.91400   035 
0.91295  8.1837 
0.91190   640 
0.91086    443 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log. 

d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d. 

Log.  Tan  Nat. 

f 

83C 


7° 


I 

Nat.  Sin  Log.  d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

12187  9-08589 
216  9.08692 
245  9.08795 
274  9.08897 
302  9.08999 

103 

103 

102 
102 
102 
101 
IO2 
IOI 
IOI 
100 
IOI 
100 
100 

99 

100 

99 
99 
98 

99 
98 
98 
98 
98 
97 
97 
97 

9l 
96 

97 
96 
96 

9I 
96 

95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
92 
92 
92 

9i 
92 

9i 
9i 
90 

9i 
90 

9i 
90 

99255  9-99675 
251  9.99674 
248  9.99672 
244  9.99670 
240  9.99669 

12278  9.08914 
308  9.09019 
338  9.09123 
367  9.09227 
397  9.09330 

105 
104 
104 
103 
104 
103 
103 

102 
I03 
102 
102 
IOI 
102 
IOI 
IOI 
IOI 
IOI 
IOO 
IOO 
IOO 
IOO 

99 
99 
99 
99 
99 
98 
98 
98 
98 

97 
98 

97 
97 
96 

9l 
96 

96 
96 

96 
95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 
93 
9i 
92 

0.91086  8.1443 
0.90981   248 
0.90877   054 
0-90773  8.0860 
0.90670   667 

60 

59 
58 

i 

5 

6 

9 

12331  9.09101 
360  9.09202 
389  9.09304 
418  9.09405 
447  9.09506 

99237  9.99667 
233  9.99666 
230  9.99664 
226  9.99663 

222  9.99661 

12426  9.09434 

456  9-09537 
485  9.09640 
515  9.09742 
544  9.09845 

0.90566  8.0476 
0.90463   285 
0.90360   095 
0.90258  7.9906 
0.90155   718 

55 

54 
53 
52 
5i 

10 

it 

12 
13 
14 

12476  9.09606 
504  9.09707 
533  9-09807 
562  9.09907 
591  9.10006 

99219  9.99659 
215  9.99658 
211  9.99656 
208  9.99655 
204  9.99653 

12574  9.09947 
603  9.10049 
633  9.10150 
662  9.10252 
692  9-10353 

0-90053  7-9530 
0.89951    344 
0.89850    158 
0.89748  7.8973 
0.89647    789 

50 

49 
48 
47 
46 

15 

16 

3 

19 

12620  9.10106 
649  9.10205 
678  9.10304 
706  9.10402 
735  9-10501 

99200  9.99651 

193  9.99648 
I89  9.99647 

186  9.99645 

12722  9.10454 
751  9.10555 
781  9.10656 
8  10  9.10756 
840  9.10856 

0.89546  7.8606 
0.89445    424 
0.89344    243 
0.89244    062 
0.89144  7.7882 

45 

44 
43 
42 
41 
40 

9 

11 

35 

34 
33 
32 
31 

20 

21 
22 

23 
24 

12764  9.10599 
793  9.10697 
822  9.10795 
851  9.10893 
880  9.10990 

99182  9.99643 
178  9.99642 
175  9.99640 
171  9-99638 
167  9.99637 

12869  9-10956 
899  9.11056 
929  9.11155 
958  9.11254 
988  9.11353 

0.89044  7.7704 

0.88746    171 
0.88647  7-6996 

25 

26 
27 
28 
29 

12908  9.11087 
937  9.11184 
966  9.11281 

995  9-II377 
13024  9.11474 

99163  9.99635 
160  9-99633 
156  9.99632 
152  9.99630 
148  9.99629 

13017  9.11452 
047  9.11551 
076  9.11649 
106  9."747 
136  9.11845 

0.88548  7.6821 
0.88449    647 
0.88351    473 
0.88253    301 
0.88155    129 

30 

31 
32 
33 

34 

13053  9.11570 
081  9.11666 
no  9.11761 
139  9-11857 
168  9.11952 

99144  9.99627 
141  9.99625 
137  9.99624 
*33  9-99622 
129  9.99620 

13165  9.11943 
195  9.12040 
224  9.12138 
254  9.12235 
284  9.12332 

0.87862    618 
0.87765    449 
0.87668    281 

30 

29 
28 
27 
26 

35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 

47 
48 

49 

13197  9.12047 
226  9.12142 
254  9.12236 
283  9.12331 
312  9.12423 

99125  9.99618 

122  9.99617 

118  9-996I5 

114  9-996I3 

no  9.99612 

13313  9.12428 
343  9.12525 
372  9.12621 
402  9.12717 
432  9.12813 

0.87572  7.5113 
0.87475  7-4947 
0-87379    78i 
0.87283    615 
0.87187    451 

25 

24 
23 

22 
21 

13341  9.12519 
370  9.12612 
399  9.12706 
427  9-^2799 
456  9.12892 

13485  9.12985 
514  9.13078 

543  9W71 
572  9.13263 

600  9-13355 

99106  9.99610 
102  9.99608 
098  9.99607 
094  9-99605; 
091  9.99603 

13461  9.12909 
491  9.13004 
52i  9.13099 
550  9.13194 
580  9.13289 

0.87091  7.4287 
0.86996    124 
0.86901  7.3962 
0.86806    800 
0.86711    639 

20 

19 

18 

17 
16 

15 

J4 
J3 

12 
II 

10 

9 
8 

i 

99087  9.99601 
083  9.99600 
°79  9-99598 
°75  9-99596 
°7i  9-99595 

13609  9.13384 
639  9.13478 
669  9.13573 
698  9.13667 
728  9.13761 

0.86616  7.3479 
0.86522    319 
0.86427    160 

0-86333     002 

0.86239  7.2844 

50 

5i 

52 
53 
54 

13629  9.13447 
658  9-13539 
687  9.13630 
716  9.13722 
744  9-13813 

99067  9.99593 
063  9-99591 
°59  9-99589 
°55  9-99588 
051  9-99586 

13758  9-I3854 
787  9.13948 
817  9.14041 
846  9.14134 
876  9.14227 

0.86146  7.2687 
0.86052   531 
0.85959    375 

0.85866     220 

0-85773    °°6 

55 

56 

% 

$> 

13773  9-I3904 
802  9.13994 
831  9.14085 
860  9.14175 
889  9.14266 
917  9.14356 

99047  9.99584 
043  9-99582 
039  9-99581 
035  9-99579 
031  999577 
027  9-99575 

13906  9.14320 
935  9-I44i2 
965  9.14504 
995  9.14597 
14024  9.14688 
054  9.14780 

0.85680  7.I9I2 

0-85588    759 
0.85496    607 
0-85403    455 
0.85312    304 
0.85220    154 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d.  Log.  Tan  Nat. 

f 

82° 


8C 


f 

Nat.  S'\n  Log.  d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

13917  9-I4356 
946  9.14445 

975  9-14535 
14004  9.14624 
033  9.14714 

89 
90 
89 
90 

89 

88 
89 
89 
88 
88 
88 
88 

87 
88 

87 
87 
87 

87 
86 
86 
87 
86 

85 
86 

85 
86 
85 
85 
85 
84 

85 
84 
84 
84 

84 

83 
84 

83 
83 
83 
83 
83 
82 
82 

83 
82 
81 
82 
82 
81 
81 
81 
81 
81 
81 
80 
80 
80 
80 
80 

99027  9.99575 
023  9-99574 
oJ9  9-99572 
OI5  9-99570 
on  9.99568 

14054  9.14780 

084  9.14872 

113  9.14963 
143  9.15054 
173  9-15145 

92 
9i 
9i 
91 
9i 
9i 
90 

9i 
90 
90 
89 
90 

89 
90 

89 
89 
88 
89 
88 
88 
88 
88 
88 
87 
88 

87 
87 
87 
86 
87 
86 
86 
86 
86 
86 
85 
86 
85 
85 
85 

85 
84 

85 
84 
84 
84 
84 

83 
84 

83 
83 
83 
83 
83 
83 
82 
82 
82 
82 
82 

0.85220  7.1154 
0.85128   004 
0-85037  7.0855 
0.84946   706 
0.84855   558 

60 

59 
58 

i 

5 

6 

7 
8 

9 

14061  9.14803 
090  9.14891 
119  9.14980 
148  9.15069 
177  9-I5I57 

99006  9.99566 
002  9.99565 
98998  9.99563 
994  9-9956i 
990  9.99559 

14202  9.15236 
232  9-I5327 
262  9.15417 
291  9.15508 
321  9.15598 

0.84764  7.0410 
0.84673   264 
0.84583    117 
0.84492  6.9972 
0.84402   827 

55 

54 
53 
52 
5i 
50 
49 
48 
47 
46 

10 

ii 

12 
13 
14 

14205  9.15245 
234  ^SSSS 

263  9.15421 
292  9.15508 
320  9.15596 

98986  9.99557 
982  9.99556 
978  9-99554 
973  9-99552 
969  9-99550 

14351  9.15688 
381  9.I5V77 
410  9.15867 
440  9.15956 
470  9.16046 

0.84312  6.9682 
0-84223   538 
0.84133    395 
0.84044    252 
0.83954    no 

15 

16 

17 
18 

19 

14349  9.15683 
378  9.15770 
407  9.15857 
436  9.15944 
464  9.16030 

98965  9.99548 
961  9.99546 
957  9-99545 
953  9-99543 
948  9.99541 

14499  9.16135 
529  9.16224 
559  9-16312 
S88  9.16401 
61  8  9.16489 

0.83865  6.8969 
0.83776    828 
0.83688    687 

0.83599    548 
0.83511    408 

45 

44 
43 
42 

41 

20 

21 

22 

23 

24 

25 

26 
27 
28 
29 

14493  9.16116 
522  9.16203 
551  9.16289 
580  9.16374 

608  9.16460 

98944  9.99539 
940  9.99537 

93i  9-99533 
927  9-99532 

14648  9.16577 
678  9.16665 
707  9-16753 
737  9.16841 
767  9.16928 

0.83423  6.8269 

0.83335    131 
0.83247  6.7994 
0.83159    856 
0.83072    720 

40 

39 
38 

1 

14637  9.16545 

666  9.16631 
695  9.16716 
723  9.16801 
752  9.16886 

98923  9.99530 
919  9.99528 
914  9-99526 
910  9.99524 
906  9.99522 

14796  9.17016 
826  9.17103 
856  9.17190 
886  9.17277 
915  9-17363 

0.82984  6.7584 
0.82897    448 
0.82810    313 
0.82723    179 
0.82637   045 

35 

34 
33 
32 
31 

30 

31 
32 
33 
34 

14781  9.16970 
8  10  9.17055 
838  9.17139 
867  9.17223 
896  9.17397 

98902  9.99520 
897  9-995*8 
893  9.99517 
889  9.99515 
884  9.99513 

14945  9.17450 
975  9-J7536 
15005  9.17622 
034  9.17708 
064  9.17794 

0.82550  6.6912 
0.82464    779 
0.82378    646 
0.82292    514 
0.82206    383 

30 

29 

28 

5 

35 

36 
37 
38 
39 
40 

41 

42 

43 
44 

14925  9.17391 
954  9.17474 
982  9.17558 
15011  9.17641 
040  9.17724 

98880  9.99511 
876  9-99509 
871  9.99507 
867  9.99505 
863  9.99503 

15094  9.17880 
124  9.17965 
153  9.18051 
183  9.18136 
213  9.18221 

0.82120  6.6252 

0.82035     122 
0.81949  6.5992 
0.8l864     863 

0.81779    734 

25 

24 

23 

22 
21 

15069  9.17807 
097  9.17890 
126  9.17973 
155  9.18055 
184  9.18137 

98858  9.99501 
854  9.99499 
849  9.99497 
845  9-99495 
841  9-99494 

15243  9.18306 
272  9.18391 
302  9.18475 
332  9.18560 
362  9.18644 

0.81694  6.5606 
0.81609    478 
0-81525    350 
0.81440    223 
0.81356    097 

20 

19 
18 
17 
16 

45. 

46 
47 
48 
49 

15212  9.18220 
241  9.18302 
270  9.18383 
299  9.18465 
327  9.18547 

98836  9.99492 
832  9.99490 
827  9.99488 
823  9.99486 
818  9.99484 

15391  9.18728 
421  9.18812 
451  9.18896 
481  9.18979 
511  9-19063 

0.81272  6.4971 
0.81188    846 
0.81104    721 
0.81021    596 
0.80937    472 

15 

14 
13 

12 
II 

TO 

9 

8 

7 
6 

50 

5i 
52 
53 
54 

15356  9.18628 
385  9.18709 
414  9.18790 
442  9.18871 
471  9.18952 

98814  9.99482 
809  9.99480 
805  9-99478 
800  9.99476 
796  9-99474 

15540  9.19146 
570  9.19229 
600  9.19312 
630  9.19395 
660  9.19478 

0.80854  6.4348 
0.80771    225 
0.80688    103 
0.80605  6.3980 
0.80522    859 

55 

56 
57 
58 

ft 

15500  9.19033 
529  9.19113 
557  9.19193 
586  9.19273 
615  9-I9353 
643  9-19433 

98791  9.99472 
787  9.99470 
782  9.99468 
778  9.99466 

773  9.99464 
769  9.99462 

15689  9.19561 
719  9.19643 
749  9.19725 
779  9.19807 
809  9.19889 
838  9.19971 

0.80439  6.3737 
0.80357    617 
0.80275    496 
0.80193    376 
0.801  1  1    257 
0.80029    138 

5 

4 
3 

2 
I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log.j  c.d.  |Log.  Tan  Nat. 

t 

sr 


9° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

15643  9.19433 
672  9.19513 
701  9.19592 
730  9-19672 
758  9.I975I 

80 

79 
80 

79 
79 
79 
79 
79 
78 
78 

79 
78 
78 
77 
78 
78 
77 
77 
77 
77 
77 
77 
76 

77 
76 

7^ 
76 

76 

75 
76 

75 
76 

75 
75 
75 
74 
75 
75 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 
73 
73 
73 
73 
72 

73 

72 
72 
73 
7i 
72 
72 

98769  9.99462 
764  9.99460 
760  9.99458 
755  9-99456 
751  9-99454 

15838  9.19971 
868  9.20053 
898  9.20134 
928  9.20216 
958  9.20297 

82 
81 
82 
81 
81 
81 
81 
81 
80 
81 
80 
80 
80 
80 
80 

79 
80 

79 
79 
79 
79 
79 
78 
79 
78 
78 
78 
78 
78 
78 

77 
78 

77 
77 
77 
77 
77 
76 

77 

76 
76 

% 

76 

75 
76 

75 
76 

75 
75 
75 
75 
75 
74 
75 
74 
75 
74 
74 
74 

0.80029  6.3138 
0.79947    oI9 
0.79866  6.2901 
0.79784    783 
0-79703   666 

60 

II 
i 

5 

6 

i 

9 

15787  9.19830 
816  9.19909 
845  9.19988 
873  9.20067 
902  9.20145 

98746  9.99452 
741  9.99450 
737  9-99448 
732  9.99446 
728  9.99444 

15988  9.20378 
16017  9.20459 
047  9.20540 
077  9.20621 
107  9.20701 

0.79622  6.2549 
0.79541   432 
0.79460   316 
0-79379   200 
0.79299   085 

55 

54 
53 
52 
51 
50 

49 

48 

47 

46 

10 

ii 

12 
13 
14 

15931  9.20223 
959  9.20302 
988  9.20380 
16017  9.20458 
046  9.20535 

98723  9-99442 
718  9.99440 
714  9.99438 
709  9.99436 
704  9-99434 

16137  9.20782 
167  9.20862 
196  9.20942 
226  9.21022 
256  9.21102 

0.79218  6.1970 
0.79138   856 
0.79058   742 
0.78978   628 
0.78898   515 

15 

16 

17 

18 

19 

16074  9-20613 
103  9.20691 
132  9.20768 
160  9.20845 
189  9.20922 

98700  9.99432 
695  9.99429 
690  9.99427 
686  9.99425 
68  1  9.99423 

16286  9.21182 
316  9.21261 
346  9.21341 
376  9.21420 
405  9.21499 

0.78818  6.1402 
0.78739   290 
0.78659   178 
0.78580   066 
0.78501  6.0955 

45 

44 
43 
42 

4i 

20 

21 
22 
23 
24 

25 

26 

27 
28 

29 

16218  9.20999 
246  9.21076 
275  9-2II53 
304  9.21229 
333  9.21306 

98676  9.99421 
671  9.99419 
667  9.99417 
662  9.99415 
657  9.99413 

16435  9-2I578 
465  9.21657 
495  9-2I736 
525  9.21814 
555  9.21893 

0.78422  6.0844 

0.78343    734 
0.78264    624 
0.78186    514 
0.78107    405 

40 

it 
% 

16361  9.21382 
390  9.21458 
419  9.21534 
447  9.21610 
476  9.21685 

98652  9-994" 
648  9.99409 
643  9.99407 
638  9.99404 
633  9-99402 

16585  9.21971 
615  9.22049 
645  9.22127 
674  9.22205 
704  9.22283 

0.78029  6.0296 

0-77951   l88 
0.77873   080 
0-77795  5-9972 
0.77717   865 

35 

34 
33 
S2 
31 

30 

3i 
32 
33 

34 

I6505  9-21761 
533  9.21836 
562  9.21912 
591  9.21987 
620  9.22062 

98629  9.99400 
624  9.99398 
619  9.99396 
614  9-99394 
609  9.99392 

16734  9.22361 
764  9.22438 
794  9.22516 
824  9.22593 
854  9.22670 

0.77639  5.9758 
0.77562   651 
0.77484   545 
0.77407    439 
0.77330    333 

30 

11 
% 

35 

36 

P 

39 

16648  9.22137 
677  9.22211 
706  9.22286 
734  9.22361 
763  9.22435 

98604  9.99390 
600  9.99388 
595  9.99385 
590  9.99383 
585  9.99381 

16884  9.22747 
914  9.22824 
944  9.22901 
974  9,22977 
17004  9.23054 

0.77253  5-9228 
0.77176    124 
0.77099    019 
0.77023  5-89I5 
0.76946    811 

25 

24 
23 

22 
21 

40 

4i 

42 

43 
44 
45 
46 
47 
48 
49 
50 
5i 
52 
53 
54 
55 
56 

P 

6S90 

16792  9.22509 
820  9.22583 
849  9.22657 
878  9.22731 
906  9.22805 

98580  9.99379 
575  9-99377 
570  9-99375 
565  9-99372 
561  9.99370 

17033  9.23130 
003  9.23206 
093  9.23283 
123  9-23359 
153  9-23435 

0.76870  5.8708 
0.76794    605 
0.76717    502 
0.76641    400 
0.76565    298 

20 

19 
18 

I76 
T5 

14 
13 

12 
II 

TO 

9 
8 

1 

l6935  9.22878 
964  9.22952 
992  9.23025 
17021  9.23098 
050  9-23171 

98556  9-99368 
55i  9-99366 
546  9.99364 
54i  9-99362 
536  9-99359 

17183  9.23510 
213  9.23586 
243  9.23661 
273  9-23737 
303  9.23812 

0.76490  5.8197 
0.76414    095 

0.76339  5-7994 
0.76263    894 
0.76188    794 

17078  9.23244 
107  9.233I7 
136  9.23390 
164  9.23462 

193  9-23535 

98531  9-99357 
526  9-99355 
521  9-99353 
5i6  9-99351 
511  9.99348 

17333  9-23887 
363  9.23962 
393  9.24037 
423  9.24112 
453  9.24186 

0.76113  5.7694 
0.76038    594 

0.75963    495 
0.75888    396 
0.75814    297 

17222  9.23607 
250  9.23679 
279  9.23752 
308  9.23823 
336  9.23895 
365  9.23967 

98506  9.99346 
501  9.99344 
496  9.99342 
49i  9-99340 
486  9.99337 
481  9.99335 

17483  9.24261 
513  9.24335 
543  9.24410 
573  9.24484 
603  9.24558 
633  9.24632 

0-75739  5.7199 
0-75665    101 
0.75590    004 
o-755I6  5-6906 
0.75442   809 
0.75368   713 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log.|  c.d.  |Log.Tan  Nat. 

f 

80° 


10° 


/ 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

17365  9-23967 
393  9-24039 
422  9.24110 
451  9.24181 
479  9.24253 

72 
7i 
7i 
72 
7i 
7i 
7i 
70 

7i 

70 

7i 
70 
70 
70 
70 
70 
70 

69 
70 

69 

69 
69 

69 
69 
69 
69 
68 
69 
68 
68 
68 
68 
68 
68 
68 

67 
68 
67 
67 
67 

t7 

67 

67 
67 
66 
67 
66 

67 
66 
66 
66 
66 

65 
66 

66 

! 
n 

98481  9.99335 
476  9-99333 
47i  9-99331 
466  9.99328 
461  9.99326 

2 

2 

3 

2 
2 
2 

3 
2 
2 
2 

3 
2 
2 
2 

3 
2 
2 

3 

2 
2 
2 

3 

2 
2 

3 

2 
2 

3 
2 

2 

3 
2 
2 
3 
2 

3 

2 
2 
3 
2 
2 

3 
2 

3 

2 
2 

3 
2 

3 

2 
2 

3 

2 

3 

2 

3 

2 
2 

3 
2 

17633  9-24632 
663  9.24706 

693  9-24779 
723  9-24853 

753  9.24926 

74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 

73 
72 

73 
72 
72 
72 
72 
72 

7i 
72 
7i 
72 
7i 
7i 
7i 
7i 
70 

7i 
7i 
70 
70 

7i 
70 
70 
70 
70 
69 
70 
69 

1° 
69 

69 
69 

69 

? 

69 

69 

68 

68 
68 
68 
67 
68 
68 
67 

0-75368  5-6713 
0.75294   617 
0.75221   521 
0.75147   425 
0.75074   329 

60 

11 
% 

55 

54 
53 
52 
5i 

6 

6 

7 

8 

9 

17508  9.24324 

537  9.24395 
565  9.24466 
594  9-24536 
623  9.24607 

98455  9-99324 
450  9-99322 
445  9-993I9 
440  9.99317 

435  9-993J5 

17783  9.25000 
813  9.25073 
843  9.25146 
873  9.25219 
903  9.25292 

0.75000  5.6234 
0.74927   140 
0.74854   045 
0.74781  5.5951 
0.74708   857 

10 

ii 

12 
13 
14 

iy 

16 
17 
18 
19 

17651  9.24677 
680  9.24748 
708  9.24818 
737  9.24888 
766  9.24958 

98430  9-993I3 
425  9-993io 
420  9.99308 
414  9.99306 
409  9.99304 

17933  9.25365 
963  9-25437 
993  9-255™ 
18023  9.25582 

053  9-25655 

0-74635  5-5764 
0.74563   671 
0.74490   578 
0.74418   485 
0-74345   393 

50 

49 
48! 
47 
46 
45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 

17794  9-25028 
823  9.25098 
852  9.25168 
880  9.25237 
9°9  9-25307 

98404  9.99301 
399  9-99299 
394  9-99297 
389  9.99294 
383  9.99292 

18083  9-25727 

"3  9-25799 
143  9.25871 

J73  9-25943 
203  9.26015 

0.74273  5-5301 
0.74201   209 
0.74129   118 
0-74057   026 
0.73985  5.4936 

20 

21 

22 

23 

24 

25 

26 
27 
28 
29 

17937  9-25376 
966  9.25445 

995  9-255I4 
18023  9.25583 
052  9.25652 

98378  9.99290 
373  9.99288 
368  9.99285 
362  9.99283 
357  9.99281 

18233  9.26086 
263  9.26158 
293  9.26229 
323  9.26301 
353  9-26372 

0.73914  54845 
0.73842   755 
0.73771   665 
0.73699   575 
0.73628   486 

18081  9.25721 
109  9.25790 
138  9.25858 
166  9.25927 
195  9-25995 

98352  9.99278 
347  9.99276 
341  9.99274 

336  9-9927I 
331  9.99269 

18384  9.26443 
414  9.26514 

504  9.26726 

0-73557  5-4397 
0.73486   308 
0.73415   219 

0-73345   131 
0.73274   043 

30 

31 

32 
33 
34 

35- 

36 

11 
i 

41 
42 

43 
44 

18224  9.26063 
252  9.26131 
281  9.26199 
309  9.26267 
338  9-26335 

98325  9-99267 
320  9.99264 
315  9.99262 
310  9.99260 
304  9.99257 

18534  9-26797 
564  9.26867 
594  9.26937 
624  9.27008 
654  9.27078 

0-73203  5-3955 
0.73133   868 
0-73063   781 
0.72992   694 
0.72922   607 

30 

29 

28 

27 
26 

25" 

24 
23 

22 
21 

20 

19 
18 

17 
16 

U 

14 
13 

12 
II 

TO 

9 
8 

I 

18367  9.26403 
395  9-26470 
424  9.26538 
452  9.26605 
481  9.26672 

98299  9.99255 
294  9-99252 
288  9.99250 
283  9.99248 

"9^272  9.99243 
267  9-9924I 
261  9.99238 
256  9-99236 
25°  9-99233 

18684  9.27148 
714  9.27218 
745  9.27288 

775  9-27357 
805  9.27427 

0.72852  5.3521 
0.72782   435 
0.72712   349 
0.72643   263 
0.72573   178 

18509  9.26739 
538  9.26806 
567  9.26873 
595  9.26940 
624  9.27007 

18835  9.27496- 
865  9.27566 
895  9-27635 
925  9.27704 
955  9-27773 

0.72504  5.3093 
0.72434   008 
0-72365  5.2924 
0.72296   839 
0.72227   755 

45 

46 
47 
48 

49 

18652  9.27073 
68  1  9.27140 
710  9.27206 
738  9.27273 
767  9-27339 

98245  9.99231 
240  9.99229 
234  9.99226 
229  9.99224 
223  9.99221 

18986  9.27842 
19016  9.27911 
046  9.27980 
076  9.28049 
106  9.28117 

0.72158  5.2672 
0.72089   588 
0.72020   505 
0.71951   422 
0.71883   339 

50 

5i 

52 
S3 
54 

18795  9-27405 
824  9.27471 

852  9-27537 
88  1  9.27602 
910  9.27668 

98218  9.99219 

212  9.99217 
207  9.99214 
201  9.99212 
196  9.99209 

19136  9.28186 
166  9.28254 
197  9.28323 
227  9.28391 
257  9.28459 

0.71814  5.2257 
0.71746   174 
0.71677   °92 
0.71609   on 
0.71541  5.1929 

55 

56 
57 
58 

6S90 

18938  9.27734 
967  9.27799 
995  9-27864 
19024  9.27930 
052  9.27995 
08  1  9.28060 

98l90  9.99207 
185  9.99204 
179  9.99202 
174  9.99200 

168  9.99197 
163  9.9919! 

19287  9.28527 
317  9.28595 
347  9.28662 
378  9.28730 

9-7I473  5-I848 
0.71405   767 
0.71338   686 
0.71270   606 
0.71202   526 
0.7H35   446 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.CotLog.|c.d. 

Log.TanNat. 

/ 

79( 


IT 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d.  Log.  Cot  Nat. 

0 

2 

3 
4 

19081  9.28060 
109  9.28125 
138  9.28190 
167  9.28254 
195  9-28319 

1 

64 
65 
65 
64 
64 

65 
64 

64 
64 
64 

A3 
64 

64 

63 
63 
64 

63 
63 
63 
63 
63 
62 

63 
62 

63 
62 
62 

63 
62 
62 
61 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
61 
60 
60 
60 
60 

59 
60 
60 

59 
60 

98163  9.99195 
157  9.99192 
152  9.99190 
146  9.99187 
140  9.99185 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 
3 

2 

3 

2 

3 

2 

3 
3 

2 

3 

2 

3 
3 

2 

3 

2 

3 
3 

2 

3 
3 

2 

3 
3 

2 

3 
3 

2 

3 
3 

2 

3 
3 

19438  9.28865 
468  9.28933 

498  9.29000 

529  9.29067 

559  9.29134 

68 
% 

A7 
67 

67 
67 
67 
67 

66 
67 
66 
67 
66 
66 
66 
66 
66 
66 
66 

65 
66 

A* 

65 
66 

65 
1S 

A5 

A5 
65 

64 

£ 
5 

64 

A5 
64 

64 
64 
64 

63 
64 

A3 
64 

63 
64 
63 
63 
A3 
63 
63 
63 

A3 
62 

63 
62 

A3 
62 

62 
62 

o-7I][35  5-I446 
0.71067   366 
0.71000   286 
0-70933   207 
0.70866   128 

60 

59 
58 

% 

5 

6 

7 
8 

9 
10 

ii 

12 
13 
14 

19224  9.28384 
252  9.28448 
281  9.28512 
309  9.28577 
338  9.28641 

98135  9.99182 
129  9.99180 
124  9-99I77 
118  9-99I75 

112  9.99172 

19589  9.29201 
619  9.29268 

649  9-29335 
680  9.29402 
710  9.29468 

0.70799  5-1049 
0-70732  5-0970 
0.70665   892 
0.70598   814 
0.70532   736 

55 

54 
53 
52 
5i 

19366  9.28705 

395  9-28769 
423  9.28833 
452  9.28896 
481  9.28960 

98107  9.99170 

ioi  9.99167 

096  9.99165 
090  9.99162 
084  9.99160 

19740  9.29535 
770  9.29601 
801  9.29668 
831  9.29734 
861  9.29800 

0.70465  5.0658 
0.70399   58i 
0-70332   5°4 
0.70266   427 
0.70200   350 

50 

49 

48 

47 
46 

45 

44 
43 
42 
4i 

15 

16 

17 
18 

19 

19509  9.29024 
538  9.29087 
566  9.29150 
595  9.29214 
623  9.29277 

98079  9.99157 

073  9-99*55 
067  9-99I52 
06  1  9.99150 
056  9.99147 

19891  9.29866 
921  9.29932 
952  9.29998 
982  9.30064 

20012  9.30130 

0.70134  5-0273 
0.70068   197 

O.70002     121 
0.69936     045 
0.69870  4.9969 

20 

21 
22 

23 
24 

19652  9.29340 
680  9.29403 
709  9.29466 

737  9-29529 
766  9.29591 

98050  9.99145 
044  9.99142 
039  9-99I40 
033  9-99I37 
027  9.99135 

20042  9.30195 
073  9.30261 
103  9.30326 

133  9-3039I 
164  9.30457 

0.69805  4.9894 

0.69739   8l9 
0.69674   744 
0.69609   669 
0.69543   594 

40 

P 
P 

25 

26 
27 
28 
29 

19794  9.29654 
823  9.29716 
851  9.29779 
880  9.29841 
908  9.29903 

98021  9.99132 
016  9.99130 
oio  9.99127 
004  9-99124 
97998  9.99122 

20194  9.30522 
224  9.30587 
254  9-30652 
285  9.30717 
315  9-30782 

0.69478  4.9520 
0.69413   446 
0.69348   372 
0.69283   298 
0.69218   225 

35 

34 
33 
32 
3i 

30 

3i 

32 
33 
34 

19937  9-29966 
965  9-30028 
994  9-30090 

20022  9.30151 
05I  9.30213 

97992  9.99119 
987  9.99117 
981  9.99114 
975  9-99II2 
969  9.99109 

20345  9-30846 
376  9.309II 
406  9.30975 
436  9.31040 
466  9.3II04 

0.69154  4.9152 
0.69089   078 
0.69025   006 
0.68960  4.8933 
0.68896   860 

30 

29 
28 

27 
26 

35 

36 

S 

39 

20079  9-30275 

108  9.30336 

136  9-30398 
165  9-30459 

97963  9.99106 
958  9.99104 
952  9.99101 
946  9.99099 
940  9.99096 

20497  9.31168 
527  9.31233 

557  9-31297 
588  9.31361 
618  9.31425 

0.68832  4.8788 
0.68767   716 
0.68703   644 
0.68639   573 
0-68575   501 

25 

24 
23 

22 
21 

40 

41 
42 
43 
44 
45" 
46 
47 
48 

49 

2O222  9.30582 
250  9.30643 

279  9-30704 
307  9.30765 
336  9.30826 

97934  9-99093 
928  9.99091 
922  9.99088 
916  9.99086 
910  9-99083 

20648  9.31489 
679  9-3I552 
709  9.31616 

739  9-3J679 
770  9.31743 

0.68511  4.8430 
0.68448   359 
0.68384   288 
0.68321   218 
0.68257   147 

20 

19 
18 

17 
16 

15 

14 
13 

12 
II 

20364  9-30887 

393  9-30947 
421  9.31008 
450  9.31068 
478  9.31129 

97905  9.99080 
899  9.99078 
893  9.99075 
887  9.99072 
88  1  9.99070 

20800  9.31806 
830  9.31870 
861  9.31933 
891  9.31996 
921  9.32059 

0.68194  4.8077 
0.68130   007 
0.68067  4-7937 
0.68004   867 
0.67941   798 

50 

51 

52 
53 
54 

20507  9.31189 
535  9-31250 
563  9-3i3I0 
592  9.31370 
620  9.31430 

97875  9-99067 
869  9.99064 
863  9.99062 

857  9-99059 
851  9.99056 

20952  9.32122 
982  9.32185 
21013  9.32248 
043  9.32311 
073  9-32373 

0.67878  4.7729 
0.67.835   659 

0-67752   59i 
0.67689   522 
0.67627   453 

10 

9 
8 

7 
6 

5 

4 
3 

2 

I 

0 

55 

56 

11 
ift 

20649  9-3I49<> 
677  9-3I549 
706  9.31609 
734  9.31669 
763  9.31728 
791  9-3I788 

97845  9-99054 
839  9-9905I 
833  9-99048 
827  9.99046 
821  9.99043 
815  9.99040 

21104  9-32436 
134  9.32498 
164  9.32561 
195  9.32623 
225  9.32685 
256  932747 

0-67564  4-7385 
0.67502   317 
0-67439   249 
0-67377   181 
0.67315   114 
0.67253   046 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

C.d. 

Log.TanNat. 

t 

78° 


12° 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

•2 

3 
4 

20791  9.31788 
820  9.31847 
848  9.31907 
877  9-3J966 
90S  9-32025 

59 

60 

59 
59 
59 
59 
59 
59 
58 
59 
59 
58 
58 
59 
58 
58 
58 
58 
58 
58 
58 

P 

57 
58 
57 
57 
58 
57 
57 
57 
56 
57 
57 
57 
56 

1 

56 
57 

^ 
56 

56 
56 
56 
56 

II 
55 
56 

ii 

55 
55 
55 
55 
55 
55 
55 
55 

97815  9.99040 
809  9.99038 
803  9.99035 
797  9-99032 
791  9.99030 

2 

3 
3 

2 

3 
3 

2 

3 

3 
3 

2 

3 
3 
3 

2 

3 
3 

3 

2 

3 
3 
3 

2 

3 
3 
3 

2 

3 
3 
3 
3 

2 

3 
3 
3 
3 
3 

2 

3 
3 
3 
3 
3 

2 

3 
3 
3 
3 
3 
3 
3 

2 

3 
3 
3 
3 
3 
3 
3 
3 

21256  9.32747 
286  9.32810 

316  9.32872 

63 
62 
61 
62 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
60 
61 
60 
60 
60 
60 
60 

59 
60 
60 
59 
60 

59 
59 
59 
60 

59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 

0.67253  4.7046 
0.67190  4.6979 
0.67128   912 
0.67067   845 
0.67005   779 

60 

59 
58 

i 

5 

6 

7 
8 

9 

20933  9-32084 
962  9.32143 
990  9.32202 
21019  9.32261 
047  9.32319 

97784  9-99027 
778  9-99024 
772  9.99022 
766  9.99019 
760  9.99016 

21408  9.33057 

438  9-33"9 
469  9-33I8o 
499  9-33242 
529  9-33303 

0.66943  4-6712 
0.66881   646 
0.66820   580 
0.66758   514 
0.66697   448 

55 

54 
53 
52 
SJ 

10 

ii 

12 
13 
M 

21076  9.32378 
104  9.32437 
132  9.32495 

161  9-32553 
189  9.32612 

97754  9-990I3 
748  9.99011 
742  9.99008 
735  9-99005 
729  9.99002 

21560  9.33365 
590  9.33426 
621  9.33487 
651  9.33548 
682  9.33609 

0.66635  4-6382 
0.66574   317 
0.66513   252 
0.66452   187 

0.66391    122 

50 

49 
48 

42 
46 

15 

16 

3 

19 

21218  9.32670 
246  9.32728 
275  9-32786 
303  9.32844 
331  9.32902 

97723  9-99000 
717  9.98997 
711  9-98994 
705  9.98991 
698  9.98989 

21712  9.33670 
743  9-33731 
773  9-33792 
804  9-33853 
834  9.33913 

0.66330  4.6057 
0.66269  4-5993 
0.66208    928 
0.66147    864 
0.66087    800 

45 

44 
43 
42 
4i 

20 

21 
22 

23 
24 

25 

26 
27 
28 
29 
30~ 
3i 
32 
33 
34 

21360  9.32960 
388  9.33018 
4i7  9-33075 
445  9-33I33 
474  9-33I90 

97692  9.98986 
686  9.98983 
680  9.98980 

673  9-98978 
667  9.98975 

21864  9-33974 
895  9-34034 
925  9.34095 
956  9.34155 
986  9.34215 

0.66026  4.5736 
0.65966    673 
0.65905    609 
0-65845    546 
0-65785    483 

40 

1 

21502  9.33248 
530  9-33305 
559  9-33362 
587  9-33420 
616  9.33477 

97661  9.98972 

655  9-98969 
648  9.98967 
642  9.98964 
636  9.98961 

22017  9-34276 
°47  9-34336 
°78  9.34396 
108  9.34456 
*39  9-345I6 

0.65724  4.5420 

0.65664   357 
0.65604   294 
0.65544   232 
0.65484   169 

35 

34 
33 
32 
3i 

21644  9-33534 
672  9-33591 
701  9.33647 
729  9-33704 
758  9-3376i 

97630  9.98958 

623  9.98955 
617  9.98953 
611  9.98950 
604  9.98947 

22169  9-34576 
200  9.34635 
231  9.34695 
261  9.34755 
292  9.34814 

0.65424  4.5107 
0.65365   045 
0-6530$  4.4983 
0.65245   922 
0.65186   860 

30 

29 

28 

3 

35 

36 
37 
38 
39 

21786  9.33818 
814  9-33874 
843  9-33931 
871  9-33987 
899  9.34043 

97598  9-98944 
592  9.98941 
585  9.98938 
579  9-98936 
573  9.98933 

22322  9.34874 
353  9-34933 
383  9-34992 
4H  9.35051 
444  9-35111 

0.65126  44799 
0.65067   737 
0.65008   676 
0.64949   615 
0.64889   555 

25 

24 
23 

22 
21 

40 

41 
42 

43 
44 

21928  9.34100 
956  9.34156 
985  9-34212 
22013  9.34268 
041  9-34324 

97566  9.98930 
560  9.98927 

553  9.98924 
547  9.98921 

541  9-98919 

22475  9-35I7o 
505  9.35229 
536  9.35288 
567  9-35347 
597  9.35405 

0.64830  4.4494 
0.64771   434 
0.64712   373 
0-64653   313 
0-64595   253 

20 

19 
18 

17 
16 

15 

14 
13 

12 
II 

45 

46 

47 
48 

49 

22070  9.34380 
098  9.34436 
126  9.34491 

155  9-34547 
183  9.34602 

97534  9.98916 
528  9.98913 
521  9.98910 
515  9.98907 
508  9.98904 

22628  9.35464 
658  9.35523 
689  9.35581 
719  9-35640 
750  9-35698 

0.64536  4.4194 
0.64477   134 
0.64419   075 
0.64360   015 
0.64302  4.3956 

50 

5i 
52 
53 
54 

22212  9.34658 
240  9.34713 
268  9.34769 
297  9.34824 
325  9-34879 

97502  9.98901 
496  9.98898 
489  9.98896 
483  9.98893 
476  9.98890 

22781  9.35757 
811  9.35815 
842  9.35873 
872  9.35931 
903  9.35989 

0.64243  4.3897' 
0.64185   838 
0.64127   779 
0.64069   721 
0.64011   662 

10 

9 
8 

7 
6 

55 

56 

% 
& 

22353  9-34934 
382  9.34989 
410  9.35044 
438  9.35099 
467  9-35I54 
495  9-35209 

97470  9.98887 
463  9.98884 
457  9.98881 
450  9.98878 
444  9.98875 
437  9.98872 

22934  9.36047 
964  9.36105 
995  9.36163 
23026  9.36221 
056  9.36279 
087  9-36336 

0-63953  4-36°4 

0^63837   488 

0.63779   430 
0.63721   372 
0.63664   315 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.JLog.TanNat. 

/ 

13° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

Hr 

6 

I 

9 

22495  9-35209 
523  9-35263 
552  9-353I8 
58o  9-35373 
608  9.35427 

54 
55 
55 
54 
54 
55 
54 
54 
54 
54 
54 
54 
54 
54 
54 
53 
54 
53 
54 
53 
53 
53 
54 
53 
53 
53 
52 
53 
53 
53 
S2 
53 
52 
52 
53 
52 
S2 
52 
52 
52 
S2 
52 
52 
52 
Si 
52 
5i 
52 
5i 
52 
51 
5i 
51 
5i 
51 
5i 
5i 
5i 
5i 

51 

97437  9-98872 
430  9.98869 
424  9.98867 
417  9.98864 
411  9.98861 

3 

2 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
3 
3 
4 

23087  9.36336 
117  9-36394 
148  9.36452 

179  9-36509 
209  9.36566 

58 
58 
57 

57 
58 
57 
57 
57 
57 
57 
57 
57 
57 
57 
56 

P 

i 

57 
56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 
56 
55 
55 
56 
55 
55 
56 
55 
55 
55 
55 
55 
54 
55 
55 
54 
55 
55 
54 
54 
55 
54 

$ 

54 

54 
54 
54 
54 

0.63664  4.3315 

0.63548   200 
0.63491   143 
0.63434   086 

60 

59 
58 
57 
56 

22637  9-3548i 
665  9-35536 

97404  9.98858 
398  9.98855 
391  9.98852 
384  9.98849 
378  9.98846 

23240  9.36624 
271  9.36681 
301  9.36738 

0.63376  4.3029 
0.63319  4-2972 
0.63262   916 
0.63205   859 
0.63148   803 

55 

54 
53 
52 
5i 

10 

II 

12 
13 
14 

15 

16 

17 
18 

19 

22778  9-35752 
807  9.35806 
835  9-35860 
863  9.35914 
892  9-35968 

97371  9-98843 
365  9.98840 
358  9.98837 
351  9.98834 
345  9-98831 

23393  9-36909 
424  9.36966 

455  937023 
485  9-37080 
516  9.37137 

0.63091  4-2747 
0.63034   691 
0.62977   635 
0.62920   580 
0.62863   524 

50 

49 
48 

47 
46 

45 

44 
43 
42 
41 

22920  9.36022 
948  9.36075 
977  9-36I29 
23005  9-36182 
033  9-36236 

97338  9.98828 
331  9.98825 
325  9.98822 
318  9.98819 
311  9.98816 

23547  9-37I93 
578  9-37250 
608  9.37306 
639  9-37363 
670  9-374I9 

0.62807  4.2468 
0.62750   413 
0.62694   358 
0.62637   303 
0.62581   248 

20 

21 
22 
23 

24 

23062  9.36289 
090  9.36342 
118  9.36395 
146  9.36449 
175  9.36502 

97304  9.98813 
298  9.98810 
291  9.98807 
284  9.98804 
278  9.98801 

23700  9.37476 
73i  9-37532 
762  9.37588 
793  9.37644 
823  9-37700 

0.62524  4.2193 
0.62468   139 
0.62412   084 
0.62356   030 
0.62300  4.1976 

40 

P 

37 
36 
~35 
34 
33 
S2 
3i 
80 
29 
28 

22 
26 

26 

24 
23 

22 
21 

25 

26 
27 
28 
29 

80 

3i 
32 
33 
34 

23203  9.36555 
231  9.36608 
260  9.36660 
288  9.36713 
316  9.36766 

97271  9.98798 
264  9.98795 
257  9-98792 
251  9.98789 
244  9.98786 

23854  9.37756 
885  9.378I2 
916  9.37868 
946  9.37924 
977  9.37980 

0.62244  4.1922 
0.62188   868 
0.62132   814 
0.62076   760 
0.62020   706 

23345  9-368I9 
373  9-3687I 
401  9.36924 
429  9.36976 
458  9.37028 

97237  9.98783 
230  9.98780 
223  9.98777 
217  9.98774 

210  9.98771 

24008  9.38035 
039  9-38091 
069  9.38147 
loo  9.38202 
131  9.38257 

0.61965  4.1653 
0.61909   600 
0.61853   547 
0.61798   493 
0.61743   441 

35 

36 

% 

39 

23486  9.37081 
514  9.37I33 
542  9.37185 
57i  9-37237 
599  9.37289 

97203  9.98768 
I96  9-98765 
189  9.98762 
182  9.98759 
176  9.98756 

24162  9.38313 
193  9-38368 
223  9-38423 
254  9-38479 
285  9-38534 

0.61687  4.1388 
0.61632   335 
0.61577   282 
0.61521   230 
0.61466   178 

40 

41 
42 
43 
44 
45 
46 

% 

49 

23627  9-37341 
656  9-37393 
684  9.37445 
712  9.37497 
740  9.37549 

97169  9.98753 
162  9.98750 
155  9-98746 
148  9.98743 
141  9.98740 

24316  9.38589 
347  9-38644 
377  9-38699 
408  9.38754 
439  9-38808 

0.61411  4.1126 
0.61356   074 

0.6I30I     022 
O.6I246  4.0970 
0.6II92    918 

20 

19 
18 

i 

23769  9-376oo 
797  9-37652 
825  9-37703 
853  9-37755 
882  9.37806 

97134  9-98737 
I27  9.98734 
120  9.98731 
H3  9.98728 

106  9-98725 

24470  9.38863 
501  9.38918 
532  9-38972 
562  9.39027 
593  9-39082 

O.6II37  4.0867 
0.61082    815 
0.61028    764 
0.60973    713 
0.609l8    662 

15 

14 
13 

12 
II 

50 

5i 
52 
53 
54 
55 
56 

% 
11 

23910  9.37858 

938  9-37909 
966  9.37960 
995  9.38011 
24023  9.38062 

97100  9.98722 
093  9.98719 
086  9.98715 
079  9.98712 
072  9.98709 

24624  9.39136 
655  9-39I90 
686  9.39245 
717  9.39299 
747  9.39353 

0.60864  4.06II 
0.60810    560 
0.60755    509 

0.60701   459 
0.60647   408 

10 

I 

-5 

4 
3 

2 

I 

0 

24051  9.38113 
079  9.38164 
108  9.38215 
136  9.38266 
164  9.38317 
192  9.38368 

97065  9.98706 
058  9.98703 
051  9.98700 
044  9.98697 
037  9.98694 
030  9.98690 

24778  9.39407 
809  9.39461 
840  9.395I5 
871  9.39569 
902  9.39623 

933  9-39677 

0-60593  4.0358 
0.60539   308 
0.60485   257 
0.60431   207 
0.60377   X58 
0.60323   108 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.dJLog.TanNat. 

/ 

76 


14C 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d.|Log.CotNat. 

0 

I 

2 

3 

"T 

6 

7 
8 

9 

24192  9.38368 
220  9.38418 
249  9.38469 
277  9-38SI9 
305  9-38570 

50 

5i 
So 
Si 
5° 
So 
5i 
5° 
So 
So 
50 
50 
5° 
So 
So 
49 
50 
5° 
49 
50 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 

49 
48 

49 
48 

49 
48 
49 
48 
48 
48 
48 
48 
48 
48 
48 
48 

47 
48 
48 

47 
48 

47 
48 
47 
47 
47 
48 

97030  9.98690 
023  9.98687 
015  9.98684 
008  9.98681 
ooi  9.98678 

3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
4 
3 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
4 

24933  9-39677 
964  9-39731 
995  9-39785 
25026  9.39838 

056  9-39892 

54 
54 
53 
54 
53 
54 
53 
54 
53 
53 
54 
53 
53 
53 
53 
53 
53 
52 
53 
53 
53 
S2 
53 
52 
53 
52 
52 
52 
53 
52 
52 
52 
52 
52 
52 
S2 
Si 
52 
52 
5i 
52 
5i 
52 
5i 
5i 
52 
5i 
5i 
5i 
5i 
5i 
5i 
5i 
5i 
5i 
51 
5° 
5i 
5i 
50 

0.60323  4.0108 
0.60269   058 
0.60215   009 
0.60162  3.9959 
0.60108   910 

60 

59 
58 

% 

24333  9-38620 
362  9.38670 
390  9.38721 
418  9.38771 
446  9.38821 

96994  9-98675 
987  9.98671 
980  9.98668 
973  9-98665 
966  9.98662 

25087  9-39945 
118  9.39999 
149  9.40052 
180  9.40106 

211  9.40159 

0.60055  3-986i 
0.60001   812 
0.59948   763 

0.59841   665 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

24474  9.38871 
503  9-3892i 
53i  9-3897I 
559  9-39021 
587  9.39071 

96959  9-98659 
952  9.98656 
945  9-98652 
937  9-98649 
930  9.98646- 

25242  9.40212 

273  9.40266 
304  9.40319 

335  9-40372 
366  9.40425 

0-59788  3-9617 
0-59734   568 
0.59681   520 
0.59628   471 
0-59575   423 

50 

49 
48 

42 
46 

45 

44 
43 

42 

4i 

15 

16 

I? 

18 

19 

24615  9.39121 
644  9.39170 
672  9-39220 
700  9.39270 
728  9-393I9 

96923  9.98643 
916  9.98640 
909  9.98636 
902  9.98633 
894  9.98630 

25397  9-40478 
428  9.40531 

459  9-40584 
490  9.40636 
521  9.40689 

0.59522  3.9375 

0-59469   327 
0.59416   279 
0-59364   232 
0.59311   184 

20 

21 
22 
23 
24 

24756  9-39369 
784  9.39418 

813  9-39467 
841  9.39517 

869  9-39566 

96887  9.98627 
880  9.98623 
873  9.98620 
866  9.98617 
858  9.98614 

25552  9.40742 
583  9.40795 
614  9.40847 
645  9.40900 
676  9.40952 

0-59258  3-9136 
0.59205   089 
0-59*53   042 
0.59100  3.8995 
0.59048   947 

40 

39 
38 
37 
36 
35~ 
34 
33 
32 
3i 

25 

26 

27 
28 
29 

24897  9-396I5 
925  9-39664 
954  9-397I3 
982  9.39762 
25010  9.39811 

96851  9.98610 
844  9.98607 
837  9.98604 
829  9.98601 
822  9.98597 

25707  9-4I005 
738  9.41057 
769  9.41109 
800  9.41161 
831  9.41214 

0.58995  3-8900 
0-58943   854 

0-58839   760 
0.58786   714 

30 

3i 
32 
33 
34 

25038  9.39860 
066  9.39909 
094  9-39958 

122  940006 
151  940055 

96815  9.98594 
807  9.98591 
800  9.98588 

793  9-98584 
786  9.98581 

25862  9.41266 
893  9-4i3I8 
924  9.41370 
955  9-41422 
986  9.41474 

0.58734  3-8667 
0.58682   621 
0.58630   575 
0-58578   528 
0.58526   482 

30 

29 
28 
27 
26 

35 

36 

P 

39 
40 

41 
42 

43 
44 

25179  940103 
207  940152 
235  940200 
263  9.40249 
29I  9.40297 

96778  9.98578 
771  9.98574 

764  9-9857I 
756  9.98568 

749  9-98565 

26017  9.41526 
048  9.41578 
079  9.41629 
no  9.41681 
141  941733 

0.58474  3.8436 
0.58422   391 

0.58371   345 
0.58319   299 
0.58267   254 

25 

24 
23 

22 
21 

25320  9.40346 
348  940394 
376  9.40442 
404  9.40490 
432  9-40538 

96742  9.98561 
734  9-98558 
727  9-98555 
719  9-98551 
712  9.98548 

26172  9.41784 
203  9.41836 
235  9.41887 
266  9.41939 
297  9.41990 

0.58216  3.8208 
0.58164   163 
0.58113   118 
0.58061   073 

0.58010    02« 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 
50~ 

5i 
52 
53 
54 
55 
56 

P 

18 

25460  9.40586 
488  9.40634 
5l6  9.40682 

545  9-40730 
573  9-40778 

96705  9.98545 
697  9-9854I 
690  9.98538 
682  9.98535 
675  9-9853I 

26328  9.42041 
359  9.42093 
390  9.42144 
421  9.42195 
452  9.42246 

0-57959  3-7983 
0.57907   938 
0.57856   893 
0.57805   848 
0.57754   804 

15 

*4 
i3 

12 
II 

10^ 

9 
8 

I 

25601  9.40825 
629  9.40873 
657  9-40921 
685  9.40968 
713  9.41016 

96667  9.98528 
660  9.98525 
653  9-9852I 
645  9-985I8 
638  9-98515 

26483  9.42297 
515  9.42348 
546  9.42399 
577  9.42450 
608  9.42501 

0.57703  3-7760 
0.57652   715 
0.57601   671 
0-57550   627 
0-57499   583 

25741  9.41063 
769  9.41111 
798  9.41158 
826  9.41205 
854  9.41252 
882  9.41300 

96630  9.98511 
623  9.98508 
615  9.98505 
608  9.98501 
600  9.98498 
593  9-98494 

26639  942552 
670  9.42603 
701  9.42653 

733  942704 
764  9.42755 
795  9.42805 

0.57448  3.7539 
0-57397   495 
0-57347   45i 
0.57296   408 

0.57245   364 
0.57195   32i 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d. 

Log.  Tan  Nat. 

t 

75C 


15C 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

~ir 

6 

7 
8 

9 

25882  9.41300 
910  9.41347 
938  9.41394 
966  9.41441 

994  9.41488 

47 
47 
47 
47 
47 
47 
46 

47 
47 
46 

47 
46 

47 
46 

47 
46 
46 

47 
46 

46 
46 
46 
46 
46 
45 

4^ 
46 
46 

45 
46 

45 
46 

$ 

45 

45 
46 

45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 

96593  9-98494 
585  9.98491 
578  9.98488 
570  9.98484 
562  9.98481 

3 

3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
4 
3 
4 

26795  942805 
826  942856 

857  942906 

888  9.42957 
920  943007 

5i 
50 
5i 
50 
So 
5i 
5° 
50 
So 
So 
5o 
50 
5° 
50 
So 
49 
50 
50 
49 
5° 
49 
So 
49 
50 
49 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 
48 

49 
49 
48 
49 
48 
48 

49 
48 
48 
48 

49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 

o-57J95  3-7321 
0.57144       277 
0.57094       234 
0.57043       191 
0.56993       J48 

60 

59 
58 

% 

55 

54 
53 
52 
5i 

26022  9.41535 
050  9.41582 
079  941628 
107  9.41675 
135  9.41722 

96555  9-98477 
547  9.98474 
540  9.98471 
532  9.98467 
524  9.98464 

26951  943057 
982  943108 
27013  943158 
044  943208 
076  9.43258 

0.56943  3.7105 
0.56892       062 
0.56842       019 
0.56792  3.6976 
0.56742       933 

10 

ii 

12 
13 
14 

26163  9.41768 
191  9.41815 
219  9.41861 
247  9.41908 
275  9.41954 

96517  9.98460 
509  9-98457 
502  9.98453 
494  9-98450 
486  9.98447 

27107  943308 

138  9-43358 
169  943408 

201    943458 

232  943508 

0.56692  3.6891 
0.56642       848 
0.56592       806 
0.56542       764 
0.56492       722 

50 

49 
48 
47 
46 

15 

16 

17 
18 

19 

26303  9.42001 
331   9.42047 

359  942093 
387  9.42140 
415  9.42186 

96479  9-98443 
471   9.98440 
463   9.98436 

456  9-98433 
448   9.98429 

27263  943558 
294  943607 
326  943657 

357  9-43707 
388  943756 

0.56442  3.6680 
0.56393       638 
0.56343       596 
0.56293       554 
0.56244       512 

45 

44 
43 

42 
41 

20 

21 
22 

23 
24 

26443  9.42232 
471  942278 
500  942324 
528  9.42370 
556  942416 

96440  9.98426 
433  9.98422 
425  9-98419 
417  9.98415 
410  9.98412 

27419  943806 
451  943855 
482  943905 

5J3  9-43954 
545  9.44004 

0.56194  3.6470 
0.5614$       429 
0.56095       387 
0.56046       346 
0.55996       305 

40 

39 
38 
37 
36 
^ 
34 
33 
32 
3i 

25 

26 
27 
28 
29 

26584  942461 
612  942507 
640  942553 
668   942599 
696  942644 

96402  9.98409 
394  9.98405 
386  9.98402 
379  9-98398 
371  9.98395 

27576  9.44053 
607  9.44102 
638  9.44151 
670  9.44201 
701  9.44250 

0-55947  3-6264 

0.55898           222 

0.55849       181 
0-55799       J40 
0-5575°       I0° 

30 

31 
32 
33 
34 

26724  9.42690 
752  942735 
780  942781 
808   942826 
836  942872 

96363  9.98391 
355  9-98388 
347  9-98384 
340  9.98381 

332  9-98377 

27732  9.44299 
764  9.44348 

795  9-44397 
826  9.44446 
858  9.44495 

0-55701  3-6059 
0.55652       01  8 
0-55603  3-5978 
0-55554       937 
0.55505       897 

30 

29 
28 

27 
26 

35 

36 

39 

26864  942917 
892  942962 
920  943008 
948   9.43053 
976  943098 

96324  9.98373 
316  9.98370 
308  9.98366 
301  9-98363 
293  9.98359 

27889  9.44544 
921  9.44592 
952  9.44641 
983  9.44690 
28015  944738 

0.55456  3-5856 
0.55408       816 
0-55359       776 
o-SSSJO       736 
0.55262       696 

25 

24 
23 

22 
21 

20 

19 
18 

17 
16 

15 

14 
13 

12 
II 

10 

9 
8 

1 

40 

41 
42 

43 
44 

27004  943143 
032  943188 
060  943233 
088   9.43278 
116  943323 

96285  9.98356 
277  9-98352 
269  9.98349 
261   9.98345 
253  9-98342 

28046  9.44787 
077  9.44836 
109  9.44884 
140  9.44933 
172  9.44981 

0.55213  3-5656 
0.55164       616 
0.55116       576 
0.55067       536 
0.55019       497 

45 

46 

2 

49 

27144  943367 
172  943412 
200  943457 
228  943502 
256  943546 

96246  9.98338 
238  9.98334 
230  9-98331 

222    9.98327 
214    9.98324 

28203  945029 
234  9-45078 
266  945126 
297  945174 
329  945222 

0.54971  3.5457 
0.54922       418 
0.54874       379 
0.54826       339 
0.54778       300 

50 

5i 

S2 
53 
54 

27284  943591 
312  9.43635 
340  943680 
368  943724 
396  943769 

96206    9.98320 

198  9-983J7 
190  9-98313 
182  9.98309 
174  9-98306 

28360  945271 
39i  9453X9 
423  9-45367 
454  945415 
486  945463 

0.54729  3.5261 

0.54681            222 

0-54633       l83 

0.54585           144 

0-54537       I05 

55 

^6 

H 

60 

27424  943813 
452  943857 
480  943901 
508  943946 
536  943990 
564  9.44034 

96166  9.98302 
158  9.98299 
150  9.98295 
142  9.98291 
134  9.98288 
126  9.98284 

28517  945511 
549  945559 
580  9.45606 
612  945654 
643  9-45702 
675  9-45750 

0.54489  3-5067 
0.54441       028 

0-54394  3-4989 
0-54346       951 
0.54298       912 
0.54250       874 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.Tan  Nat. 

/ 

74C 


r 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

i 

2 

3 
4 

27564  9.44034 
592  9.44078 
620  9.44122 
648  9.44166 
676  944210 

44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
44 
43 
43 
44 
43 
44 
43 
43 
43 
43 
43 
44 
43 
42 
43 
43 
43 
43 
43 
42 

43 
42 

43 
42 
43 
42 
43 
42 
42 
42 

43 
42 

42 
42 
42 
42 
42 
42 
4i 
42 
42 
42 

41 

42 

4i 
42 
41 
42 
4i 
42 

96126  9.98284 
118  9.98281 
no  9-98277 
102  9.98273 
094  9.98270 

3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
3 
4 
4 
4 
3 
4 
4 
4 
3 
4 
4 
4 
3 
4 
4 
4 
3 
4 
4 
4 
3 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
3 

28675  9-45750 

706  945797 
738  945845 
769  9.45892 
801  9-45940 

47 
48 

47 
48 

47 
48 

47 
48 

47 
47 
47 
48 

47 
47 
47 
47 
47 
47 
47 
46 

47 
47 
47 
46 

47 
47 
46 

% 

46 

47 

4* 
46 

47 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

45 
46 
46 
46 

45 
46 

45 
46 

45 
45 
46 
45 
45 
46 

45 

0.54250  3.4874 
0.54203       836 
0.54155       798 
0.54108       760 
0.54060       722 

60 

59 
58 
57 
56 

5 

6 

7 
8 

9 

27704  9.44253 
731  9.44297 

759  9-44341 
787  9-44385 
815  9.44428 

96086  9.98266 
078  9.98262 
070  9.98259 
062  9.98255 
054  9.98251 

28832  9.45987 
864  946035 
895  9.46082 
927  946130 
958  9.46177 

0.54013  3.4684 
0.53965       646 
0.53918       608 
0-53870       570 
0-53823       533 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

27843  9.44472 
871   9.44516 
899  9.44559 
927  9.44602 
955  9.44646 

96046  9.98248 
037  9.98244 
029  9.98240 

021     9.98237 
013    9.98233 

28990  9.46224 
29021  9.46271 

-053  9-463I9 
084  946366 
116  946413 

0.53776  3-4495 
0.53729       458 
0.53681       420 

0.53634       383 
0.53587       346 

50 

49 

48 

$ 

15 

16 

i? 
18 
19 

27983  9.44689 
28011  9.44733 
°39  9-44776 
067  9.44819 
095  9.44862 

96005    9.98229 

95997  9-98226 
989  9.98222 
981   9.98218 
972  9.98215 

29147  9.46460 
179   946507 

210    946554 
242    946601 
274    946648 

0-53540  3.4308 
0-53493       271 
0.53446       234 

0-53399       197 
0.53352       160 

45 

44 
43 
42 

41 

20 

21 
22 
23 
24 

28123  9-44905 
150  9.44948 
178  9.44992 
206  9.45035 
234  9-45077 

95964  9.98211 
956  9.98207 
948  9.98204 
940  9.98200 
931   9.98196 

29305    946694 

337   9-4674I 
368   946788 
400  9.46835 
432  9.46881 

0.53306  3.4124 
0.53259       087 
0.53212       050 
0.53165       014 
o-53«9  3-3977 

40 

39 
38 
37 
36 

25 

26 
27 
28 
29 

28262  9.45120 
290  9.45163 
318  9.45206 
346  9.45249 
374  9-45292 

95923  9-98192 
915  9.98189 
907   9.98185 
898   9.98181 
890  9.98177' 

29463  9.46928 

495   9-46975 
526  .947021 
558   947068 
590  947114 

0-53072  3-3941 
0.53025      904 
0.52979       868 
0.52932       832 
0.52886       796 

35 

34 
33 
32 
3i 
313 
29 
28 

27 
26 

25 

24 
23 

22 
21 

30 

3i 

32 
33 
34 

28402  945334 
429  9-45377 
457  9-454I9 
485  9.45462 

5i3  945504 

95882  9.98174 
874  9.98170 
865  9.98166 
857   9.98162 
849  9.98159 

29621   947160 
653   9-47207 
685   947253 
716  947299 
748   947346 

0.52840  3.3759 

0.52793       723 
0.52747        687 
0.52701        652 
0.52654       616 

35 

36 

P 

39 

28541  9.45547 
569  9-45589 
597  945632 
625  9.45674 
652  9-457I6 

95841   9.98I55 
832  9.98151 
824   9.98147 
816  9.98144 
807  9.98140 

29780  947392 
811   9-47438 
843   947484 
875   9-47530 
906  947576 

0.52608  3.3580 
0.52562        544 
0.52516       509 
0.52470       473 
0.52424       438 

40 

41 
42 
43 
44 

28680  9.45758 
708  9.45801 
736  9.45843 
764  9.45885 
792  945927 

95799  9-98136 
791   9.98132 
782  9.98129 
774  9-98125 
766  9.98121 

29938   947622 
970  947668 
30001   9-477I4 
033  9-47760 
065   947806 

0.52378  3-3402 
0.52332       367 
0.52286       332 
0.52240       297 
0.52194       261 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

28820  9.45969 
847  9.46011 
875  9.46053 
903  9.46095 
931  9.46136 

95757  9-98II7 
749  9.98113 
740  9.98110 
732  9.98106 
724  9.98102 

30097  947852 
128   947897 
160  947943 
192  947989 
224  948035 

0.52148  3.3226 
0.52103       191 
0-52057        156 

0.520II           122 
0.51965          087 

15 

14 

13 

12 
II 

50 

5i 
52 
53 
54 

28959  9.46178 
987  9.46220 
29015  9.46262 
042  9.46303 
070  9.46345 

95715  9.98098 
707  9.98094 
698  9.98090 
690  9.98087 
68  1   9.98083 

30255  948080 
287  948126 
319  948171 
351  9.48217 
382  948262 

0.51920   3.3052 
0.51874          017 
0.51829   3.2983 

0.51783       948 
0.51738       914 

10 

9 
8 

7 
6 

55 

56 
57 
58 

% 

29098  9.46386 
126  9.46428 
154  9.46469 
182  946511 
209  946552 
237  946594 

95673  9-98079 
664  9.98075 
656  9.98071 
647  9.98067 
639  9.98063 
630  9.98060 

30414  948307 

509  948443 
541  948489 
573   948534 

0-51693  3-2879 
0.51647       845 
0.51602       811 

o-5I5S7       777 
0.51511       743 
0.51466       709 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.  Tan  Nat. 

f 

73C 


17° 


/ 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

29237  9.46594 
265  9.46635 
293  9.46676 
321  9.46717 
348  9.46758 

4i 
4i 
4i 
41 
42 
4i 
41 
41 
4i 
4i 
40 
4i 
4i 
41 
41 
40 

41 
40 

41 
40 

41 
40 

4i 
40 

40 

41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 

39 
40 

40 

39 
40 
40 
39 
40 

39 
40 

39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 

9563o  9-98o6o 
622  9.98056 
613  9.98052 
605  9.98048 
596  9.98044 

4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

30573  948534 

605  9-48579 
637  948624 

669  948669 

700  948714 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
43 
43 
43 
43 
44 

£ 

0.51466  3.2709 
0.51421  675 
0.51376  641 
0.51331  607 
0.51286  573 

60 

59 
58 
57 
56 
55" 
54 
53 
S2 
5i 

5 

6 

7 

8 

9 

29376  9.46800 
404  9.46841 
432  9.46882 
460  9.46923 
487  9.46964 

95588  9.98040 
579  9-98036 
571  9.98032 
562  9.98029 
554  9.98025 

30732  948759 
764  948804 
796  9.48849 
828  948894 
860  948939 

0.51241  3.2539 
0.51196  506 
0.51151  472 
0.51106  438 
0.51061  405 

10 

ii 

12 
13 

14 

29515  947005 
543  9.47045 
571  9.47086 
599  9.47127 
626  9.47168 

95545  9-98021 
536  9.98017 
528  9.98013 
519  9.98009 
511  9-98005 

30891  948984 
923  949029 

955  9-49073 
987  949118 
31019  949163 

0.51016  3.2371 

0.50971  338 
0.50927  305 
0.50882  272 
0-50837  238 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

29654  9.47209 
682  9.47249 
710  9.47290 
737  9-47330 
765   947371 

95502  9.98001 
493  9-97997 
485  9-97993 
476  9.97989 
467  9.97986 

31051  949207 
083  949252 
115  949296 
147  949341 
178  949385 

0.50793  3.2205 
0.50748  172 
0.50704  139 
0-50659  106 
0.50615  073 

45 

44 
43 
42 

4i 

20 

21 
22 

23 
24 

29793  9474" 
821   9.47452 
849  947492 
876  947533 
904  947573 

95459  9-97982 
450  9-97978 
441  9.97974 
433   9-97970 
424  9.97966 

31210  949430 
242  949474 

274  9495I9 
306  949563 
338  949607 

0.50570  3-2041 
0.50526  008 
0.50481  3.1975 
0.50437  943 
0.50393  9io 

40 

39 
38 
37 
36 

25 

26 
27 
28 
29 
30 
3i 
32 
33 
34 
35~ 
36 

P 

39 

29932  9476i3 
960  947654 
987  947694 
36015  947734 
043  947774 

95415  9.97962 
407  9-97958 
398  9.97954 
389  9-97950 
380  9.97946 

31370  949652 
402  9.49696 
434  9-49740 
466  9.49784 
498  949828 

0.50348  3-1878 
0.50304  845 
0.50260  813 
0.50216  780 
0.50172  748 

35 

34 
33 
S2 
3i 

30071  9478i4 
098  9.47854 
126  947894 
154  947934 
182  947974 

95372  9.97942 
363  9.97938 
354  9-97934 
345  9-97930 
337  9-97926 

31530  949872 
562  9.49916 
594  9-49960 
626  9.50004 
658  9.50048 

0.50128  3.1716 
0.50084  684 
0.50040  652 
049996  620 
0-49952  588 

30 

22 
26 

"25" 

24 
23 

22 
21 

20 

19 

18 

11 

15 

14 
13 

12 
II 

30209  9.48014 
237  948054 
265  9.48094 
292  9.48133 
320  9.48173 

95328  9.97922 
319  9.97918 
310  9.97914 
301  9.97910 
293  9-97906 

31690  9.50092 
722  9.50136 
754  9-50i8o 
786  9.50223 
818   9.50267* 

049908  3.1556 
049864  524 
049820  492 
0-49777  460 
0-49733  429 

40 

4i 

42 
43 

44 

30348  948213 
376  948252 
403  948292 
431  948332 
459  948371 
30486  948411 
514  948450 
542  948490 
570  948529 
597  9.48568 

95284  9.97902 
275  9.97898 
266  9.97894 
257  9-97890 
248  9.97886 

31850  9.50311 
882  9.50355 
914  9.50398 
946   9.50442 

32010  9.50529 
042  9.50572 
074   9.50616 
106  9.50659 
139  9-50703 
32171   9.50746 
203   9.50789 
235   9-50833 
267   9.50876 
299   9.50919 

0.49689  3.1397 
049645  366 
0.49602  334 

049558  303 
049515  271 

45 

46 

47 
48 

49 
50 

5i 
52 
53 
54 
5T 
56 

3 

18 

95240  9.97882 
231  9.97878 

222    9.97874 
213    9.97870 
204    9.97866 

0.49471  3.1240 
0.49428  209 
0.49384  178 
0.49341  146 
049297  "5 

30625  9.48607 
653  9.48647 
680  948686 
708  948725 
736  948764 

95195    9.97861 

1  86   9.97857 

'77   9.97853 
168   9.97849 

159  9-97845 

0.49254  3.1084 
049211  053 

0.49167  022 
049124  3.0991 
049081  961 

10 

i 

T 

4 
3 

2 

I 

0 

30763  948803 
791  9.48842 
819  948881 
846  948920 
874  948959 
902  948998 

95150  9.97841 
142  9.97837 

133   9-97833 
124  9.97829 
115  9.97825 
106  9.97821 

32331   9.50962 
363   9.51005 
396  9.51048 
428   9.51092 
460  9-5"35 
492  9.51178 

0.49038  3.0930 
0.48995  899 

0.48952  868 
048908  838 
048865  807 
048822  777 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.|Log.TanNat. 

/ 

18C 


r 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d. 

Log.  Cot  Nat. 

0 

i 

2 

3 

4 

30902  9.48998 
929  9.49037 
957  9.49076 
985  9.49115 
31012  9.49153 

39 
39 
39 
38 
39 
39 
38 
39 
39 
38 
39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

11 

38 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
38 
37 
37 
37 
37 
37 
38 
37 
37 

P 

37 
37 
37 

P 

37 

95106  9.97821 
097  9.97817 
088  9.97812 
079  9.97808 
070  9.97804 

4 
5 
4 
4 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
5 
4 
4 
4 
.5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 

32492  9.51178 
524  9.51221 
556  9.51264 
588  9.51306 
621  9.51349 

43 
43 
42 
43 
43 
43 
43 
42 

43 
43 
42 
43 
43 
42 

43 
42 
42 

43 
42 

43 
42 
42 
42 
43 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
41 
42 
42 
42 
42 

41 

42 

4i 
42 
42 
41 
42 
41 
41 
42 

4i 
42 
4i 
41 
4i 
42 

4i 
4i 
41 
4i 
4i 

0.48822  3.0777 
0.48779   746 
0.48736   716 
0.48694   686 
0.48651   655 

60 

1 

5 

6 

9 

31040  9.49192 
068  9.49231 
095  9.49269 
123  9.49308 
151  9.49347 

95061  9.97800 
052  9-97796 
043  9-97792 
°33  9-97788 
024  9.97784 

32653  9-5J392 
685  9.51435 
717  9.51478 
749  9-5I520 
782  9.51563 

0.48608  3.0625 

0-48565   595 
0.48522   565 
0.48480   535 
0.48437   505 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

31178  9.49385 
206  9.49424 
233  949462 
261  9.49500 
289  9-49539 

95oi5  9-97779 
006  9.97775 

94997  9-97771 
988  9.97767 

979  9-97763 

32814  9.51606 
846  9.51648 
878  9.51691 
911  9-5!734 
943  9.5J776 

0.48394  3.0475 
0.48352   445 
0.48309   415 
0.48266   385 
0.48224   356 

50 

49 
48 

47 
46 

45 

44 
43 
42 
4i 
40 

I 

15 

16 

2 

19 

31316  9.49577 
344  9-496i5 
372  9.49654 
399  9.49692 
427  9.49730 

94970  9.97759 
961  9.97754 
952  997750 
943  9-97746 
933  9-97742 

32975  9-5i8i9 
33007  9.51861 
040  9.51903 
072  9.51946 
104  9.51988 

0.48181  3.0326 
0.48139   296 
0.48097   267 
0.48054   237 
0.48012   208 

20 

21 
22 

23 
24 

25 

26 

27 
28 

29 

31454  9-49768 
482  9.49806 
510  9.49844 
537  9.49882 
565  9.49920 

94924  9-97738 
9i5  9-97734 
906  9.97729 

897  9-97725 
888  9.97721 

33*36  9-52031 
169  9.52073 
201  9.52115 

233  9-52I57 
266  9.52200 

0.47969  3.0178 
0.47927   149 

0.47885    120 
0.47843    090 

0.47800   061 

3*593  949958 
620  9.49996 
648  9.50034 
675  9.50072 
703  9-Soiio 

94878  9.97717 
869  9.97713 
860  9.97708 
851  9.97704 
842  9.97700 

33298  9-52242 
330  9.52284 
363  9.52326 
395  962368 
427  9.52410 

047758  3-0032 
0.47716   003 
0.47674  2.9974 
047632   945 
0.47590   916 

35 

34 
33 
32 
31 

30 

31 
32 
33 
34 

3I73°  9-50I48 
758  9.50185 
786  9.50223 
813  9.50261 
841  9.50298 

94832  9.97696 
823  9-9769I 
814  9.97687 
805  9.97683 
795  9-97679 

33460  9.52452 
492  9.52494 
524  9.52536 
557  9.52578 
589  9.52620 

0.47548  2.9887 
0.47506   858 
0.47464   829 
0.47422   800 
047380   772 

30 

29 
28 

27 

26 

35 

36 

% 

39 
40 

41 
42 
43 
44 

31868  9.50336 
896  9-50374 
923  9-504ii 
951  9-50449 
979  9-50486 

94786  9.97674 
777  9-97670 
768  9.97666 
758  9.97662 
749  9-97657 

33621  9.52661 
654  9-52703 
686  9.52745 
718  9-52787 
751  9.52829 

047339  2-9743 
0.47297   714 
047255   686 
047213   657 
047171   629 

25 

24 
23 

22 
21 

20 

19 

18 

17 
16 

32006  9.50523 
034  9-5056i 
061  9.50598 
089  9.50635 
116  9-50673 

94740  9.97653 
73°  9-97649 
721  9.97645 
712  9.97640 
702  9.97636 

33783  9-52870 
816  9.52912 
848  9.5295? 
88  1  9.52995 
913  9.53037 

047130  2.9600 
0.47088   572 
047047   544 
0.47005   515 
0.46963   487 

45 

46 

47 
48 

49 
50 

Si 
52 
53 
54 
55 
56 

i 
$ 

32144  9.50710 
171  9.50747 

199  9-50784 
227  9.50821 
254  9-50858 

94693  9.97632 
684  9.97628 
674  9.97623 
665  9.97619 
656  9.97615 

33945  9-53078 
978  9.53120 
34010  9.53161 
043  9.53202 
075  9.53244 

046922  2.9459 
0.46880   431 
0.46839   403 
046798   375 
046756   347 

15 

14 
13 

12 
II 

32282  9.50896 
309  9-50933 
337  9-50970 
364  9.51007 
392  9.51043 

94646  9.97610 
637  9-97606 
627  9.97602 
618  9-97597 
609  9-97593 

34108  9.53285 
140  9-53327 
173  9.53368 
205  9-53409 
238  9-53450 

0.46715  2.9319 
0.46673   291 
046632   263 
046591   235 
0.46550   208 

10 

9 
8 

I 

32419  9.51080 
447  9.51117 

474  9-5"54 
502  9.51191 
529  9.51227 
557  9.51264 

94599  9-97589 
590  9-97584 
580  9.97580 

57i  9-97576 
561  9-97571 
552  9.97567 

34270  9.53492 
303  9-53533 
335  9-53574 
368  9.53615; 
400  9-53656 
433  9.53697 

0.46508  2.9180 
0.46467   152 
046426   125 
046385   097 
0.46344   070 
0.46303   042 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.  Log.Tan  Nat. 

r 

71C 


19C 


r 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.|c.d. 

Log.  Cot  Nat. 

0 

i 

2 

3 

4 

32557  9-51264 
584  9.51301 
612  9.51338 

639  9-5*374 
667  9.51411 

37 

£ 

37 
36 
37 
36 
37 
36 
36 

1 

it 

37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 

36 
36 

35 
36 

35 
36 
35 

II 

35 
35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 

94552  9.97567 
542  9.97563 
533  9.97558 
523  9-97554 
5*4  9-9755° 

4 
5 
4 
4 
5 
4 
5 
4 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 

34433  9-53697 
465  9.53738 
498  9-53779 
530  9.53820 
563  9-5386I 

4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
40 

4i 

4i 
40 

4i 
4i 
40 

4i 
40 

4i 
40 
4i 
40 

4i 
40 
40 
4i 
40 
40 
4i 
40 
40 

40 
40 
40 
40 
40 
40 
40 
40 
40 
4° 
40 
40 
39 
4° 
40 
40 
39 
40 
40 
39 
40 
39 
40 

39 
40 

39 
40 

39 
39 
40 

0.46303  2.9042 
0.46262   015 
0.46221  2.8987 
0.46180   960 
046139   933 

60 

59 
58 
57 
56 

5 

6 

7 

8 

9 

32694  9.51447 
722  9.51484 
749  9-$*S2<> 
777  9-5I557 
804  9.51593 

94504  9-97545 
495  9-97541 
485  9.97536 
476  9.97532 
466  9.97528 

34596  9.53902 
628  9.53943 
661  9.53984 
693  9-5402$ 
726  9.54065 

0.46098  2.8905 
0.46057   878 
0.46016   851 
0-45975   824 
0-45935   797 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

32832  9.51629 
859  9.51666 
887  9.51702 
914  9.51738 
942  9.51774 

94457  9.97523 
447  9.97519 
438  9-975I5 
428  9.97510 
418  9.97506 

34758  9.54io6 
791  9.54147 
824  9.54187 
856  9.54228 
889  9.54269 

0.45894  2.8770. 

0.45853   743 
0.45813   716 
0.45772   689 
0.45731   662 

50 

49 
48 

:76 

45 

44 
43 
42 
4i 

15 

16 
17 
18 
19 

32969  9.51811 
997  9.51847 
33024  9.51883 
051  9.51919 
°79  9-5I955 

94409  9.97501 

399  9-97497 
390  9.97492 
380  9.97488 
370  9.97484 

34922  9.54309 
954  9-54350 
987  9-54390 
35020  9.54431 
052  9.54471 

0.45691  2.8636 
0.45650   609 
0.45610   582 
045569   556 
045529   529 

20 

21 
22 
23 

24 

33106  9.51991 
134  9.52027 
161  9.52063 
189  9.52099 
216  9.52^5 

94361  9.97479 
351  9.97475 
342  9.97470 
332  9-97466 
322  9.97461 

35085  9.54512 
"8  9-54552 
IS°  9-54593 
l83  9.54633 
216  9.54673 

0.45488  2.8502 
0.45448   476 
0.45407   449 
045367   423 
045327   397 

40 

P 

i 

25 

26 

27 
28 
29 

33244  9.5217* 
271  9.52207 
298  9.52242 
326  9.52278 
353  9.52314 

94313  9-97457 
303  9-97453 
293  9.97448 
284  9.97444 
274  9-97439 

35248  9.54714 
281  9.54754 
314  9-54794 
346  9.54835 

0.45286  2.8370 
0.45246   344 
0.45206   318 
0.45165   291 
0.45125   265 

35 

34 
33 
32 
3i 
30 
29 
28 

22 
26 

25 

24 
23 

22 
21 

20 

19 
18 

17 
16 

"16 

J4 
13 

12 
II 

30 

3i 
32 
33 
34 
35 
36 

% 

39 
40 

41 
42 
43 
44 

33381  9.52350 
408  9.52385 
436  9.52421 
463  9.52456 
490  9.52492 

94264  9-97435 
254  9-97430 
245  9-97426 
235  9.97421 
225  9.97417 

35412  9.5491$ 
445  9-54955 
477  9-54995 
510  9.55035 

543  9-55075 

0.45085  2.8239 
0.45045   213 
0.45005   187 
0.44965   161 
0.44925   135 

33518  9.52527 
545  9-52563 
573  9-52598 
600  9.52634 
627  9.52669 

94215  9.97412 
206  9.97408 
196  9.97403 
186  9.97399 
176  9-97394 

35576  9.55115 
608  9.55155 
641  9.55195 
674  9-55235 
707  9-55275 

0.44885  2.8109 
0.44845   083 
0.44805   057 
0.44765   032 
0.44725   006 

33655  9-52705 
682  9.52740 
710  9.52775 
737  9.52811 
764  9.52846 

94167  9.97390 
157  9.97385 
147  9.9738i 
137  9-97376 
127  9.97372 

35740  9-553I5 
772  9-55355 
805  9.55395 
838  9-55434 
871  9-55474 

0.44685  2.7980 
0.44645   955 
0.44605   929 

0.44526   878 

45 

46 

47 
48 

49 

33792  9.52881 
819  9.52916 
846  9-5295I 
874  9.52986 
901  9.53021 

94118  9.97367 
108  9.97363 
°98  9.97358 
088  9-97353 
°78  9-97349 

35904  9-555I4 
937  9-55554 
969  9-55593 
36002  9.55633 

035  9-55673 

0.44486  2.7852 
0.44446   827 
0.44407   801 
0.44367   776 
0.44327   751 

50 

5i 

52 
53 
54 

33929  9.53056 
956  9.53092 
983  9.53126 
34011  9.53161 
038  9.53I96 

94068  9.97344 
058  9-97340 
°49  9-97335 
039  9-97331 
029  9.97326 

36068  9.55712 
ioi  9-55752 
134  9-55791 
l67  9-5583I 
199  9-55870 

0.44288  2.7725 
0.44248   700 
0.44209   675 
0.44169   650 
0.44130   625 

10 

9 
8 

7 
6 

55 

56 

% 

ft 

34065  9.5323I 
093  9-53266 

120  9-53301 

J47  9.53336 
175  9-53370 

202  9.53405 

94019  9.97322 
009  9.97317 
93999  9-97312 
989  9-97308 
979  9.97303 
969  9.97299 

36232  9.55910 
265  9-55949 
298  9-55989 
331  9.56028 
364  9.56067 
397  9.56107 

0.44090  2.7600 
0.44051   575 
0.44011   550 
0.43972   525 
0-43933   5oo 
0.43893   475 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.|Log.TanNat. 

/ 

70° 


20° 


r 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d.  Log.  Cot  Nat. 

0 

i 

2 

3 
4 

34202  9.53405 
229  9.53440 
257  9-53475 
284  9-53509 
3"  9-53544 

35 

35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 

93969  9.97299 
959  9-97294 
949  9.97289 
939  9-97285 
929  9.97280 

5 
5 
4 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
5 

36397  9-56107 
430  9-56146 
463  9-56185 
496  9.56224 
529  9.56264 

39 
39 
39 
40 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
38 
39 
39 
39 
38 
39 

9 

P 

39 
38 

39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

£ 

i 
38 

0.43893  2.7475 
0.43854  450 
0.43815  425 

043776  400 

0.43736  376 

60 

59 
58 

H 
55 

54 
53 
52 
5i 
50 
49 
48 
47 
46 

5 

6 

9 

34339  9-53578 
366  9.53613 

393  9.53647 
421  9.53682 
448  9.537K5 

93919  9.97276 
909  9.97271 
899  9.97266 
889  9.97262 
879  9.97257 

36562  9.56303 
595  9-56342 
628  9.56381 
661  9.56420 
694  9-56459 

043697  2.7351 
0.43658  326 

0.43619  302 

0.43580  277 
043541  253 

10 

ii 

12 
13 
14 

15 

16 

17 
18 

19 

•34475  9.53751 
503  9-53785 
53°  9.53819 
557  9-53854 
584  9.53888 

93869  9-97252 
859  9.97248 
849  9.97243 
839  9-97238 
829  9.97234 

36727  9-56498 
760  9.56537 
793  9.56576 
826  9.56615 

859  9-56654 

043502  2.7228 
043463  204 
043424  179 

0.43385  155 
043346  130 

34612  9.53922 
639  9-53957 
666  9.53991 
694  9.54025 
721  9.54059 

93819  9.97229 
809  9.97224 
799  9-97220 
789  9.97215 
779  9-97210 

36892  9.56693 
925  9-56732 
958  9.5677I 
991  9.56810 
37024  9.56849 

0.43307  2.7106 
043268  082 
043229  058 

0.43190  034 
043151  009 

45 

44 
43 
42 
4i 

20 

21 
22 

23 

24 

25 

26 

29 

34748  9.54093 
775  9.54127 
803  9-54161 
830  9.54195; 
857  9-54229 

93769  9.97206 
759  9-97201 
748  9.97196 
738  9.97192 
728  9.97187 

37057  9.56887 
090  9.56926 
123  9.56965 
157  9.57004 
190  9.57042 

043113  2.6985 
0.43074  961 

043035  937 
0.42996  913 
0.42958  889 

40 

P 
1 

34884  9.54263 
912  9.54297 

939  9-54331 
966  9.54365 

993  9-54399 

937i8  9.97182 
708  9.97178 
698  9-97I73 
688  9.97168 
677  9.97163 

37223  9.57081 
256  9.57120 
289  9.57158 
322  9.57197 
355  9.57235 

042919  2.6865 
0.42880  841 
042842  818 
0.42803  794 
0.42765  770 

35 

34 
33 
32 
31 

30 

3i 
32 
33 

34 

35021  9.54433 
048  9.54466 
°75  9.54500 
102  9.54534 
130  9-54567 

93667  9.97159 
657  9.97154 
647  9-97I49 
637  9.97I45 
626  9.97140 

4 
5 
5 
4 
5 
5 
5 
4 
5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 

37388  9.57274 
422  9-57312 
455  9-57351 
488  9.57389 
521  9.57428 

0.42726  2.6746 
0.42688  723 
0.42649  699 
042611  675 
042572  652 

30 

29 

28 

% 

35 

36 
37 
38 
39 
40 
41 
42 
43 
44 

35*57  9-546oi 
184  9.54635 
211  9.54668 
239  9-54702 
266  9.54735 

93616  9.97135 
606  9-97130 
596  9.97126 
585  9.97121 
575  9-97"6 

37554  9.57466 
588  9.57504 
621  9-57543 
654  9.5758I 
687  9.57619 

0.42534  2.6628 
042496  605 
042457  581 
0.42419  558 
042381  534 

25 

24 
23 

22 
21 

20 

19 

18 

3 

35293  9.54769 
320  9.54802 

347  9.54836 
375  9.54869 
402  9.54903 

93565  9.97«i 
555  9-97107 
544  9.97102 

534  9.97097 
524  9.97092 

37720  9-57658 
754  9-57696 
787  9-57734 
820  9.57772 
853  9-578io 

0.42342  2.6511 
0.42304  488 
042266  464 
0.42228  441 
0.42190  418 

45 

46 

% 

49 

35429  9.54936 
456  9.54969 
484  9.55003 
5ii  9.55036 
538  9-55069 

93514  9.97087 
503  9.97083 
493  9.97078 
483  9-97073 
472  9.97068 

37887  9.57849 
920  9.57887 

953  9-57925 
986  9.57963 
38020  9.58001 

042151  2.6395 
0.42113  371 
042075  348 
042037  325 
041999  302 

15 

14 
13 

12 
II 

50 

5i 

52 
53 
54 

35565  9-55I02 
592  9.55136 
619  9.55169 
647  9-55202 
674  9.55235 

93462  9.97063 
452  9.97059 
441  9.97054 
431  9.97049 
420  9.97044 

38053  9.58039 
086  9.58077 

120  9.58115 
153  9-58153 

186  9.58191 

0.41961  2.6279 
041923  256 
041885  233 

0.41847  210 
0.41809  187 

10 

I 
I 

55 

56 
57 
58 

6S90 

36701  9.55268 
728  9-55301 
755  9-55334 
782  9.55367 
810  9-55400 
837  9-55433 

93410  9.97039 
400  9.97035 
389  9.97030 
379  9.97025 
368  9.97020 
358  9-97015 

38220  9.58229 
253  9.58267 
286  9.58304 
320  9.58342 
353  9-58380 
386  9.58418 

041771  2.6165 

o-4I733  J42 
0.41696  119 
0.41658  096 
041620  074 
041582  051 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.Log.TanNat. 

/ 

69 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 
5 
6 

9 

35837  9-55433 
864  9.55466 
891  9.55499 
918  9-S5532 
945  9-55564 

33 
33 
33 
32 
33 
33 
33 
32 
33 
33 
32 
33 
S2 
33 
32 
33 
32 
33 
32 
32 
33 
S2 
32 
33 
32 
32 
32 
S2 
32 
33 
32 
S2 
32 
32 
32 
31 
32 
32 
32 
32 
32 
3i 
32 
32 
32 
3i 
32 
3i 
32 
32 
3i 
S2 
3i 
3i 
32 

H 

3i 
3i 
32 

93358  9.97015 
348  9.97010 
337  9.97005 
327  9.97001 
316  9.96996 

5 
5 
4 
5 
5 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

38386  9.58418 
420  9.58455 

453  9-58493 
487  9-5853I 
520  9.58569 

37 

38 
38 
38 
37 
38 

P 

38 
37 
38 

% 

37 
37 
38 
37 
38 
37 
37 
37 
38 
37 
37 
37 
37 
38 
37 
37 
37 
37 
37 
37 
37 
37 
37 

32 
36 

37 
37 
37 

% 

37 
37 

11 

37 
37 
36 

% 
% 

37 
36 

% 

11 

0.41582  2.6051 
0.41545   028 
0.41507   006 
0.41469  2.5983 
0.41431   961 

60 

P 

fa 

35973  9-55597 
36000  9.55630 
027  9.55663 
054  9-55695 
08  1  9.55728 

93306  9.96991 
295  9-96986 
285  9.96981 
274  9.96976 
264  9-9697I 

38553  9-586o6 
587  9.58644 
620  9.58681 
654  9-587I9 
687  9.58757 

0.41394  2.5938 
0.41356   916 
0.41319   893 
0.41281   871 
0.41243   848 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

36108  9.55761 

135  9-55793 
162  9.55826 

190  9-55858 
217  9-5589I 

93253  9.96966 
243  9.96962 
232  9-96957 

222  9.96952 
211  9.96947 

38721  9.58794 
754  9-58832 
787  9.58869 
821  9.58907 
854  9.58944 

0.41206  2.5826 
0.41168   804 
0.41131   782 
0.41093   759 
0.41056   737 

50 

49 

48 

47 
46 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 

20 

19 
18 

% 

IT 

14 
13 

12 
II 

To 
1 

15 

16 

17 
18 

19 

w 

21 
22 

23 

24 

36244  9.55923 
271  9-55956 
298  9-55988 
325  9.56021 

93201  9.96942 
I90  9-96937 

180  996932 

I69  9.96927 
I59  9.96922 

38888  9.58981 
921  9.59019 
955  9-59056 
988  9.59094 
39022  9.59131 

0.41019  2.5715 
0.40981   693 
0.40944   671 
0.40906   649 
0.40869   627 

36379  9-56085 
406  9.56118 
434  9-56150 
461  9.56182 
488  9.56215 

93148  9.96917 
137  9.96912 
127  9.96907 

39055  9-59I68 
089  969205 
122  9.59243 
156  9.59280 
190  9-593I7 

0.40832  2.5605 
0.40795   583 
040757   56i 
0.40720   539 
0.40683   517 

25 

26 
27 
28 
29 

36515  9-56247 
542  9.56279 

569  9-56311 
596  9-56343 
623  9-56375 

93°95  9-96893 
084  9.96888 
074  9.96883 
063  9.96878 
052  9.96873 

39223  9.59354 
257  9-59391 
290  9.59429 
324  9.59466 

357  9.59503 

0.40646  2.5495 
0.40609   473 
0.40571   452 
040534   430 
0.40497   408 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

36650  9.56408 
677  9-56440 
704  9.56472 

731  9-56504 
758  9-56536 

93042  9.96868 
03I  9.96863 
020  9.96858 

oio  9.96853 

92999  9.96848 

39391  9.59540 
425  9-59577 
458  9-59614 
492  9.59651 
526  9.59688 

0.40460  2.5386 
0.40423   365 
0.40386   343 
0.40349   322 
0.40312   300 

36785  9-56568 
812  9-56599 
839  9-56631 
867  9.56663 
894  9-56695 

92988  9.96843 
978  9.96838 
967  9.96833 
956  9.96828 

945  9.96823 

39559  9-59725 
593  9.59762 
626  9.59799 
660  9.59835 
694  9.59872 

0.40275  2.5279 
040238   257 
0.40201   236 
0.40165   214 
0.40128   193 

40 

41 
42 

43 
44 

36921  9.56727 
948  9.56759 
975  9-56790 
37002  9.56822 
029  9.56854 

92935  9.96818 
924  9.96813 

9o2  9'968o8 

39727  9.59909 
761  9.59946 

795  9.59983 
829  9.60019 
862  9.60056 

0.40091  2.5172 
040054   150 
040017   129 
0.39981   108 
0.39944   086 

45 

46 

47 
48 

49 

37056  9.56886 
083  9-569I7 
no  9.56949 
137  9-56980 
164  9.57012 

92881  9.96793 
870  9.96788 

859  9-96783 
849  9.96778 
838  9-96772 

39896  9.60093 
930  9.60130 
963  9.60166 
997  9.60203 
40031  9.60240 

0.39907  2.5065 
0.39870   044 
0.39834   023 

0.39797    O02 

0.39760  2.4981 

50 

51 

S2 
53 
54 

37I9I  9-57044 
218  9.57075 
245  9-57I07 
272  9-57J38 
299  9-57169 

92827  9.96767 

816  9.96762 
805  9-96757 
794  9-96752 
784  9.96747 

92773  9.96742 
762  9.96737 

751  9.96732 
740  9.96727 
729  9.96722 
718  9.96717 

40065  9.60276 
098  9.60313 
132  9.60349 
166  9.60386 
200  9.60422 

0.39724  2.4960 

0.39687   939 
0.39651   9i8 
0.39614   897 
0.39578   876 

55 

56 

% 
& 

37326  9.57201 
353  9-57232 
380  9.57264 
407  9-57295 
434  9-57326 
461  9.57358 

40234  9.60459 
267  9.60495 
301  9.60532 
335  9.60568 
369  9.60605 
403  9.60641 

0.39541  2.4855 
0.39505   834 
0.39468   813 
0-39432   792 
0-39395   772 
0-39359   751 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.Tan  Nat 

f 

68 


22° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat 

0 

I 

2 

3 
4 
~5~ 

6 

I 

9 

37461  9.57358 
488  9-57389 
515  9.57420 
542  9-57451 
569  9-57482 

3i 
3i 
3i 
3i 
32 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
30 
3i 
3i 
3i 
3° 
3i 
3i 
3i 
30 
3i 
30 
3i 
30 
3i 
30 
3i 
30 
3i 
3° 
3i 
30 
30 
30 
3i 
3° 
30 
30 
3i 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
3° 

92718  9.96717 
707  9.96711 
697  9.96706 
686  9.96701 
675  9.96696 

6 
5 

5 
5 
5 
5 
5 
5 
6 

5 
5 
5 
5 
5 

6 
5 
5 
5 
5 
6 
5 
5 
5 
5 
6 
5 
5 
5 

6 
5 
5 
5 
6 

5 
5 

5 

5 
5 
5 
6 
5 

5 

5 
5 
6 

5 
5 
6 

5 
5 

40403  9.60641 
436  9.60677 
470  9.60714 
504  9.60750 
538  9.60786 

3* 

1 

36 
37 
36 
36 
36 
36 
37 

% 
36 

36 
36 
36 

% 
36 

36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 
35 
36 
36 
36 
35 
36 

36 
35 
36 
35 
36 

P 

35 
36 
35 

35 

35 
35 
35 
36 
35 
35 
35 
35 

0-39359  2.4751 
0.39323   730 
0.39286   709 
0.39250   689 
0.39214   668 

60 

ft 

57 
56 
55 

54 
53 
52 
Si 

37595  9-575I4 
622  9-57545 
649  9-57576 
676  9.57607 
703  9-57638 

92664  9.96691 

642  9.96681 
631  9.96676 
620  9.96670 

40572  9.60823 
606  9.60859 
640  9.60895 
674  9.60931 
707  9.60967 

°'39irT7  2.4648 
0.39141   627 
0.39105   606 
0.39069   586 
0.39033   566 

10 

ii 

12 

J3 
14 
T5~ 

16 
17 
18 
19 

37730  9-57669 
757  9-57700 
784  9-57731 
811  9.57762 

838  9-57793 

92609  9.96665 
598  9.96660 
587  9.96655 
576  9.96650 
565  9.96645 

40741  9.61004 
775  9.61040 
809  9.61076 
843  9.61112 
877  9.61148 

0.38996  2.4545 
0.38960   525 
0.38924   504 
0.38888   484 
0.38852   464 

50 

49 
48 
47 
46 

45 

44 
43 
42 
4i 

37865  9-57824 
892  9-57855 
919  9.57885 
946  9-579I6 
973  9-57947 

92554  9.96640 
543  9-96634 
532  9.96629 
521  9.96624 
510  9.96619 

40911  9.61184 
945  9.61220 
979  9.61256 
41013  9.61292 
047  9.61328 

0.38816  2.4443 
0.38780   423 
0.38744   403 
0.38708   383 
0.38672   362 

20 

21 
22 

23 

24 

37999  9-57978 
38026  9.58008 
053  9-58039 
080  9.58070 
107  9.58101 

92499  9.96614 
488  9.96608 
477  9.96603 
466  9.96598 
455  9-96593 

41081  9.61364 
115  9.61400 
149  9.61436 
183  9.61472 
217  9.61508 

0.38636  2.4342 
0.38600   322 
0.38564   302 
0.38528   282 
0.38492   262 

40 

9 

% 

25 

26 

27 
28 
29 
30 
3i 
32 
33 
34 
35 
36 

39 

38134  9.58131 
161  9.58162 
188  9.58192 
215  9.58223 
241  9.58253 

92444  9.96588 
432  9-96582 
421  9.96577 
410  9.96572 
399  9-96567 

41251  .9.61544 

285  9.61579 
319  9.61615 
353  9.61651 
387  9.61687 

0.38456  2.4242 

0.38421    222 
0.38385    202 
0.38349    182 
0.38313    162 

35 

34 
33 
S2 
3i 

38268  9.58284 

295  9-583I4 
322  9.58345 

349  9-58375 
376  9.58406 

92388  9.96562 
377  9.96556 
366  9.96551 
355  9-0546 
343  9-9654I 

41421  9.61722 

455  9-6I758 
490  9.61794 
524  9.61830 
558  9.61865 

0.38278  2.4142 
0.38242    122 
0.38206    102 
0.38170    083 
0.38135    063 

30 

% 
5 

38403  9.58436 
43°  9.58467 
456  9.58497 

483  9-58527 
510  9.58557 

92332  9.96535 
321  9.96530 
310  9.96525 
299  9-9652o 
287  9.96514 

41592  9.61901 
626  9.61936 
660  9.61972 
694  9.62008 
728  9.62043 

0.38099  2.4043 
0.38064    023 
0.38028    004 
0.37992  2.3984 
0.37957    964 

25 

24 
23 

22 
21 

40 

41 
42 

43 
44 

38537  9-58588 
564  9.58618 
59i  9.58648 
617  9.58678 
644  9.58709 

92276  9.96509 
265  9-96504 
254  9-96498 
243  9.96493 
231  9.96488 

41763  9.62079 
797  9.62114 
831  9.62150 
865  9.62185 
899  9.62221 

0.37921  2.3945 
0.37886    925 
0.37850    906 

0.37815   886 
0-37779   867 

20 

*9 
18 

11 

45 

46 

47 
48 

49 

38671  9.58739 
698  9.58769 

725  9-58799 
752  9.58829 
778  9.58859 

92220  9.96483 
209  9.96477 
198  9.96472 
186  9.96467 
175  9.96461 

41933  9-62256 
968  9.62292 
42002  9.62327 
036  9.62362 
070  9.62398 

0.37744  2.3847 
0.37708   828 
0-37673   808 
0-37638   789 
0.37602   770 

15 

*4 
J3 

12 
II 

50 

5i 

52 
53 
54 

38805  9.58889 
832  9.58919 
859  9-58949 
886  9.58979 
912  9.59009 

92164  9.96456 
152  9.96451 
141  9.96445 
130  9.96440 
119  9.96435 

42105  9.62433 
139  9.62468 
173  9.62504 
207  9.62539 
242  9.62574 

0.37567  2.3750 
0.37532   73i 
0.37496   712 
0.37461   693 
0.37426   673 

10 

1 

7 

6 

"T 

4 
3 

2 

I 

0 

55 

56 

% 
& 

38939  9-59039 
966  9.59069 

993  9-59098 
39020  9.59128 
046  9.59158 
073  9.59188 

92107  9.96429 
096  9.96424 
085  9.96419 

073  9-964I3 
062  9.96408 
050  9.96403 

42276  9.62609 
310  9.62645 
345  9.62680 
379  9.62715 
413  9.62750 
447  9-62785 

0-37391  2.3654 
0.37355   635 
0.37320   616 

0-37285   597 
0-37250   578 
0.37215   559 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.|Log.TanNat. 

/ 

67° 


r 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat  Tan  Log. 

c.d.  Log.  Cot  Nat. 

0 

i 

•2 

3 
4 
~5" 

6 

7 
8 

9 

39073  9-59i88 
100  9.59218 
127  9.59247 

*53  9-59277 
180  9.59307 

30 
29 
30 
30 
29 

30 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
29 

30 
29 

29 
29 

29 
30 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
28 
29 

29 
29 
28 

29 
29 
29 
28 
29 
28 
29 
29 
28 
29 
28 

29 
28 
28 

92050  9.96403 
039  9.96397 
028  9.96392 
016  9.96387 
°°5  9-96381 

6 
5 
5 
6 

5 
6 
5 

5 
6 

5 

5 
6 

5 
6 

5 
5 
6 

5 
6 

5 
6 

5 
6 

I 

5 
6 

5 
6 
6 

5 
6 

5 
6 
6 

5 
6 
6 

5 
6 

5 
6 

42447  9.62785 

482  9.62820 
516  9.62855 
551  9.62890 
585  9.62926 

35 
35 

3I 
36 

35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 

0.37215  2.3559 
0.37180  539 
0.37145  520 
0.37110  501 
0.37074  483 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

39207  9-59336 
234  9-59366 
260  9.59396 
287  9-59425 
314  9-59455 

91994  9.96376 
982  9.96370 
971  9.96365 
959  9-96360 
948  9.96354 

42619  9.62961 

654  9.62996 

688  9.63031 
722  9.63066 
757  9-63101 

0.37039  2.3464 
0.37004  445 
0.36969  426 
0.36934  407 
0-36899  388 

10 

ii 

12 
13 
14 

39341  9.59484 
367  9  595H 
394  9-J9543 
421  9-59573 
448  9.59602 

91936  9.96349 
925  9.96343 
914  9-96338 
902  9-96333 
891  9.96327 

42791  9.63135 
826  9.63170 
860  9.63205 
894  9.63240 
929  9.13275 

0.36865  2.3369 
0.36830  351 

0.36795  S32 
0.36760  313 
0.36725  294 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

39474  9-59632 
501  9-59661 
528  9-59690 
555  9-59720 
581  9.59749 

91879  9.96322 
868  9.96316 
856  9.96311 
845  9-96305 
833  9.0300 

42963  9.63310 
998  9.63345 
43032  9.63379 
067  9.63414 
lox  9.63449 

0.36690  2.3276 
0.36655  257 
0.36621  238 

0.36586  220 
0.3655I  201 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 

20 

21 
22 

23 
24 

25" 

26 

3 

29 

w 

31 

32 

33 
34 

39608  9.59778 

635  9-598o8 
661  9-59837 
688  9.59866 
715  9.59895 

91822  9.96294 
8  10  9.96289 
799  9.96284 
787  9.96278 
775  9.96273 

43136  9.63484 
170  9-635x9 

239  9.63588 
274  9.63623 

0.36516  2.3183 
0.36481  164 
0.36447  146 
0.36412  127 

0.36377  I09 

39741  9.59924 
768  9-59954 
795  969983 
822  9.60012 
848  9.60041 

91764  9.96267 
752  9.96262 
741  9.96256 
729  9.96251 
718  9.96245 

43308  9.63657 
343  9-63692 
378  9-63726 
412  9.63761 
447  9.63796 

0.36343  2.3090 
0.36308  072 
0.36274  053 
0.36239  035 
0.36204  017 

35 

34 
33 
32 
3i 
30 
29 
28 

z 

39875  9.60070 
902  9.60099 
928  9.60128 

955  9.60157 
982  9.60186 

91706  9.96240 
694  9.96234 
683  9.96229 
671  9.96223 
660  9.96218 

43481  9.63830 
516  9.63865 
550  9.63899 

585  9.63934 
620  9.63968 

0.36170  2.2998 
0-36135  980 
0.36101  962 
0.36066  944 
0.36032  925 

35 

36 
37 
38 
39 
40 

4i 
42 

43 
44 

40008  9.60215 
035  9.60244 
062  9.60273 
088  9.60302 
"5  9-60331 

91648  9.96212 
636  9.96207 
625  9.96201 
613  9.96196 
601  9.96190 

43654  9.64003 
689  9.64037 
724  9.64072 
758  9.64106 
793  9.64140 

0-35997  2.2907 
0.35963  889 
0.35928  871 
0.35894  853 
0.35860  835 

25 

24 
23 

22 
21 

40141  9.60359 
168  9.60388 
195  9.60417 

221  9.60446 
248  9.60474 

91590  9.96185 
578  9-96179 
566  9.96174 
555  9.96168 
543  9-96162 

43828  9.64175 
862  9.64209 
897  9.64243 
932  9.64278 
966  9.64312 

0.35825  2.2817 
0.35791  799 
0-35757  78i 
0.35722  763 
0.35688  745 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

40275  9.60503 
301  9.60532 
328  9.60561 

355  9-60589 
381  9.60618 

91531  9.96157 
519  9.96151 
508  9.96146 
496  9.96140 
484  996135 

44001  9.64346 
036  9.64381 
071  9.64415 
105  9.64449 
140  9.64483 

0.35654  2.2727 
0-35619  709 
0.35585  691 
0-35551  673 
o-SSS1?  655 

15 

14 
13 

12 
II 

50 

5i 
52 
53 

54 

40408  9.60646 
434  9.60675 
461  9.60704 
488  9.60732 
514  9.60761 

91472  9.96129 
461  9.96123 
449  9.96118 
437  9.96112 
425  9.96107 

44175  9.64517 

210  9.64552 
244  9.64586 
279  9.64620 
314  9.64654 

0.35483  2.2637 
0.35448  620 
0.35414  602 
0.35380  584 
0.35346  566 

10 

I 

55 

56 

% 

18 

40541  9.60789 
567  9.60818 
594  9.60846 
621  9.60875 
647  9.60903 
674  9.60931 

91414  9.96101 
402  9.96095 
390  9.96090 
378  9.96084 
366  9-96079 
355  9-96073 

44349  9.64688 
384  9.64722 
418  9.64756 
453  9.64790 
488  9.64824 
523  9.64858 

0.35312  2.2549 
0-35278  53i 
0.35244  5i3 
0.35210  496 
0.35176  478 
0.35142  460 

5 

4 
3 

2 

I 

0 

|Nat.CoSLog.  d. 

Nat.  Sin  Log.  d.  |Nat.CotLog. 

c.d.|Log.TanNat. 

/ 

66( 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d.  |Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

40674  9.60931 
700  9.60960 
727  9.60988 
753  9.61016 
780  9.61045 

29 
28 
28 
29 
28 
28 
28 

29 

28 

28 
28 
28 
28 
28 
28 
28 
29 

27 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 

27 
28 
28 
27 
28 
28 

2 

27 
28 

27 
28 

27 
28 

27 

28 

27 
27 
28 
27 
27 

27 
28 
27 
27 
27 
27 
28 
27 
27 

91355  9-96073 
343  9-0o67 
331  9.96062 
319  9.96056 
307  9.96050 

6 

6 

5 
6 

5 
6 
6 

5 
6 
6 

I 

6 
6 

6 

5 
6 
6 
6 
5 
6 
6 

5 
6 
6 
6 

5 
6 
6 
6 
6 

5 
6 
6 
6 
6 

5 
6 
6 
6 
6 

5 
6 
6 
6 
6 
6 

6 
6 
6 
6 
6 
6 
5 

44523  9.64858 
558  9.64892 
593  9.64926 
627  9.64960 
662  9.64994 

34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 

0.35142  2.2460 
0.35108   443 

0.35074   425 
0.35040   408 
0.35006   390 

60 

59 
58 
57 
56 

5 

6 

7 
8 

9 
10 

ii 

12 
13 
14 

40806  9.61073 
833  9.61101 
860  9.61129 
886  9.61158 
913  9.61186 

91295  9.96045 
283  9.96039 
272  9.96034 
260  9.96028 
248  9.96022 

44697  9.65028 
732  9.65062 
767  9.65096 
802  9.65130 
837  9.65164 

0.34972  2.2373 
0.34938   355 
0.34904   338 
0.34870   320 
0.34836   303 

55 

54 
53 
52 
51 
50 

49 
48 

47 
46 

45 

44 
43 
42 
4i 
40 

P 
1 

40939  9.61214 
966  9.61242 
992  9.61270 
41019  9.61298 
045  9.61326 

91236  9.96017 
224  9.96011 

212  9.96005 
200  9.96000 

188  9.95994 

44872  9.65197 
907  9.65231 
942  9.65265 
977  9.65299 
45012  9-65333 

0.34803  2.2286 
0.34769   268 
0-34735   251 
0.34701   234 
0.34667   216 

15 

16 

S 

19 

41072  9.61354 
098  9.61382 
125  9.61411 
151  9.61438 
178  9.61466 

91176  9.95988 
164  9.95982 

152  995977 
140  9-95971 
128  9.95965 

45047  9.65366 
082  9.65400 
117  9.65434 
152  9.65467 
187  9-6550I 

0.34634  2.2199 
0.34600   182 
0.34566   165 
0-34533   148 
0-34499   130 

20 

21 
22 

23 
24 

41204  9.61494 
231  9.61522 
257  9-6I550 
284  9.61578 
310  9.61606 

91116  9.95960 
104  9.95954 
092  9.95948 
080  9.95942 
068  9-95937 

45222  9.65535 

292  9.65602 
327  9.65636 
362  9.65669 

0.34465  2.2113 
0.34432   096 
0.34398   079 
0.34364   062 
0-34331   °45 

25 

26 
27 
28 
29 

41337  9-6K334 
363  9.61662 
390  9.61689 
416  9.61717 
443  9-6I745 

91056  9.95931 
044  9.95925 
032  9-95920 
020  9.95914 
008  9.95908 

45397  9-65703 
432  9-65736 
467  9.65770 
502  9.65803 
538  9.65837 

0.34297  2.2028 
0.34264   on 
0.34230  2.1994 
0.34197   977 
0.34163   960 

35 

34 
33 
32 
3i 

30 

31 
32 
33 

34 

41469  9.61773 
496  9.61800 
522  9.61828 
549  9.61856 
575  9.61883 

90996  9.95902 
984  9.95897 
972  9.95891 
960  9.95885 
948  9.95879 

45573  9-65870 
608  9.65904 

643  9-65937 
678  9.65971 
713  9.65004 

0.34130  2.1943 
0.34096   926 
0.34063   909 
0.34029   892 
0.33996   876 

30 

29 
28 

% 

35 

36 

P 

39 

41602  9.61911 
628  9.61939 
655  9.61966 
68  1  9.61994 
707  9.62021 

90936  9-95873 
924  9.95868 
911  9.95862 
899  9.95856 
887  9.95850 

45748  9.66038 
784  9.66071 
819  9.66104 
854  9.66138 
889  9.66171 

0.33962  2.1859 
0.33929   842 
0.33896   825 
0.33862   808 
0.33829   792 

25 

24 
23 

22 
21 

40 

4i 
42 

43 

44 

41734  9.62049 
760  9.62076 
787  9.62104 
813  9.62131 
840  9.62159 

90875  9-95844 
863  9-95839 
851  9-95833 
839  9.95827 
826  9.95821 

45924  9.66204 
960  9.66238 
995  9.66271 
46030  9.66304 
065  9-66337 

0-33796  2.1775 
0.33762   758 
0.33729   742 
0.33696   725 
0.33663   708 

20 

19 
18 

;76 

45 

46 

47 
48 

49 

41866  9.62186 
892  9.62214 
919  9.62241 
945  9.62268 
972  9.62296 

90814  9.95815 
802  9.95810 
790  9-95804 
778  9-95798 
766  9.95792 

46101  9.66371 
136  9.66404 
171  9.66437 
206  9.66470 
242  9.66503 

0.33629  2.1692 
0.33506   675 
0.33563   659 
0.33530   642 
0-33497   625 

15 

14 
13 

12 
II 

50 

51 

S2 
53 
54 

41998  9.62323 
42024  9.62350 
051  9.62377 
077  9.62405 
104  9.62432 

90753  9-95786 
741  9.95780 
729  9-95775 
717  9-95769 
704  9.95763 

46277  9.66537 
312  9.66570 
348  9.66603 
383  9.66636 
418  9.66669 

0.33463  2.1609 
0-33430   592 
0-33397   576 
0.33364   56o 
0-33331   543 

10 

9 
8 

7 
6 

55 

56 
57 
58 

i90 

42130  9.62459 
156  9.62486 
183  9.62513 
209  9.62541 
235  9.62568 
262  9.62595 

90692  9.95757 
680  9.95751 
668  9-95745 
655  9-95739 
643  9-95733 
631  9.95728 

46454  9.66702 
489  9-66735 
525  9.66768 
560  9.66801 
595  9-66834 
631  9.66867 

0.33298  2.1527 
0.33265   510 
0.33232   494 
0.33199   478 
0.33166   461 
o.33J33   445 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.|Log.TanNat. 

f 

65 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2. 

3 
4 

"T 

6 
9 

42262  9.62595 
288  9.62622 
315  9.62649 
341  9.62676 
367  9.62703 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 

27 
27 

27 
26 
27 
27 
26 

27 
26 

22 

26 

27 
26 

27 
26 

27 
26 

27 
26 
26 
27 
26 
26 
27 
26 
26 
26 

27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

90631  9.95728 
618  9.95722 
606  9-957J6 
594  9-95710 
582  9.95704 

6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

5 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

I 

6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

7 
6 
6 
6 
6 
6 
6 

I 

6 

6 

46631  9.66867 
666  9.66900 
702  9.66933 
737  9.66966 
772  9.66999 

33 
33 
33 
33 
33 
33 
33 
33 
32 
33 
33 
33 
33 
32 
33 
33 
33 
32 
33 
33 
32 
33 
33 
32 
33 
32 
33 
33 
32 
33 
32 
33 
S2 
33 
32 
32 
33 
32 
33 
32 
32 
33 
38 
32 
33 
32 
32 
32 
33 
32 
32 
32 
32 
33 
S2 
32 
32 
32 
32 
32 

0.33133  2.1445 
0.33100   429 
0.33067   413 
0.33034   396 
0.33001   380 

60 

59 
58 

% 

42394  9-62730 
420  9.62757 
446  9.62784 
473  9.62811 
499  9.62838 

90569  9-95698 
557  9-95692 
545  9-95686 
532  9.95680 
520  9.95674 

46808  9.67032 
843  9-67065 
879  9.67098 
914  9.67131 
950  9.67163 

0.32968  2.1364 

0-32935   348 
0.32902   332 
0.32869   315 
0.32837   299 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

15 

16 
17 
18 
19 

42525  9.62865 
552  9.62892 
578  9.62918 
604  9.62945 
631  9.62972 

90507  9.95668 
495  9-95663 
483  9-95657 
470  9-9565I 
458  9.95645 

46985  9.67196 
47021  9.67229 
056  9.67262 
092  9.67295 
128  9.67327 

0.32804  2.1283 
0.32771   267 
0.32738   251 
0.32705   235 
0.32673   219 

50 

49 
48 

47 
46 

46 

44 
43 
42 
4i 
40 
39 
38 
37 
36 

42657  9.62999 
683  9.63026 
709  9.63052 
736  9.63079 
762  9.63106 

90446  9.95639 

433  9-95633 
421  9.95627 
408  9.95621 
396  9-956I5 

47163  9.67360 
199  9.67393 
234  9-67426 
270  9.67458 
305  9.67491 

0.32640  2.1203 
0.32607   187 
0.32574   171 
0.32542   155 
0.32509   139 

20 

21 
22 

23 

24 

42788  9.63133 

8l5  9-63I59 
841  9.63186 
867  9.63213 
894  9.63239 

90383  9-95609 
371  9.95603 
358  9-95597 
346  9-95591 
334  9-95585 

47341  9.67524 

412  9.67589 
448  9.67622 
483  9.67654 

0.32476  2.1123 
0.32444   107 
0.32411   092 
0.32378   076 
0.32346   060 

25 

26 

27 

28 
29 

42920  9.63266 
946  9.63292 
972  9-63319 
999  9.63345 
43025  9.63372 

90321  9.95579 
309  9-95573 
296  9-95567 
284  9-9556I 
27i  9-95555 

47519  9.67687 
555  9-67719 
590  9-67752 
626  9.67785 
662  9.67817 

0.32313  2.1044 
0.32281   028 
0.32248   013 
0.32215  2.0997 
0.32183   981 

35 

34 
33 
32 
3i 

30 

3i 

32 
33 
34 

43051  9.63398 
077  9.63425 
104  9.63451 
130  9.63478 

90259  9-95549 
246  9-95543 
233  9-95537 
221  9-95531 
208  9.95525 

47698  9.67850 
733  9-67882 
769  9.67915 
805  9.67947 
840  9.67980 

0.32150  2.0965 
0.32118   950 
0.32085   934 
0.32053   918 
0.32020   903 

30 

29 
28 
27 
26 

35 

36 

3 

39 
40~ 

41 
42 

43 
44 

43182  9.63531 
209  9.63557 

235  9-63583 
261  9.63610 
287  9.63636 

90196  9.95519 
183  9-955I3 
171  9-95$o7 
158  9-95500 
146  9-95494 

47876  9.68012 
912  9.68044 
948  9.68077 
984  9.68109 
48019  9.68142 

0.31988  2.0887 
0.31956   872 
0.31923   856 
0.31891   840 
0.31858   825 

25 

24 
23 

22 
21 

43313  9-63662 
340  9.63689 
366  9.63715 
392  9-6374I 
418  9.63767 

90133  9.95488 
120  9.95482 
108  9.95476 
095  9-95470 
082  9.95464 

48055  9.68174 
091  9,68206 
127  9.68239 
163  9.68271 
198  9.68303 

0.31826  2.0809 

o.3!794   794 
0.31761   778 
0.31729   763 
0.31697   748 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 
50~ 

5i 
52 

53 

54 

43445  9-63794 
471  9.63820 
497  9-63846 
523  9.63872 
549  9-63898 

90070  9.95458 
057  9-95452 
045  9-95446 
032  9.95440 
019  9-95434 

48234  9.68336 
270  9.68368 
306  9.68400 
342  9.68432 
378  9.68465 

0.31664  2.0732 
0.31632   717 
0.31600   701 
0.31568   686 
0.31535   671 

15 

14 
13 

12 
II 

To~ 

i 

43575  9-63924 
602  9.63950 
628  9.63976 
654  9.64002 
680  9.64028 

90007  9.95427 
89994  9.95421 
981  9.95415 
968  9-95409 
956  9-95403 

48414  9.68497 
450  9.68529 
486  9.68561 
521  9.68593 

48593  9-68658 
629  9.68690 
665  9.68722 
701  9.68754 
737  9-68786 
773  9.68818 

0.31503  2.0655 
0.31471   640 
0.31439   625 
0.31407   609 
0.31374   594 

55 

56 
57 
58 

§9o 

43706  9.64054 
733  9.64080 
759  9.64106 
785  9.64132 
811  9.64158 
837  9.64184 

89943  9-95397 
93o  9-95391 
918  9.95384 
905  9-95378 
892  9-95372 
879  9-95366 

0.31342  2.0579 
0.31310   564 
0.31278   549 
0.31246   533 
0.31214   518 
0.31182   503 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.  Log.  Tan  Nat. 

1 

64C 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.lcd. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

43837  9-64184 
863  9.64210 
889  9.64236 
916  9.64262 
942  9.64288 

26 
26 
26 
26 

25 
26 
26 
26 
26 

25 
26 
26 

2£ 
26 

26 

3 
11 

25 
26 

25 
26 

25 
26 

2 

25 
25 
26 

25 
25 
26 

25 
25 

25 
26 

25 
25 
25 
25 

2I 

£6 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
24 

25 
25 
25 
25 
25 

89879  9.95366 
867  9-95360 
854  9-95354 
84i  9.95348 
828  9-95341 

6 
6 
6 
7 
6 
6 
6 
6 
7 
6 
6 
6 
6 
7 
6 
6 
6 

I 

6 
6 

7 
6 
6 
6 

7 
6 
6 

7 
6 
6 
6 

7 
6 
6 

7 
6 
6 
7 
6 
6 

I 

7 
6 
6 

I 

6 

7 
6 

7 
6 
6 

7 
6 

7 
6 
6 

7 

48773  9.68818 
809  9.68850 

845  9.68882 

88  1  9.68914 

917  9.68946 

32 
S2 
32 
32 
32 
32 
S2 
32 
S2 
32 
32 
S2 
32 
32 
32 
3i 
S2 
S2 
S2 
32 
3i 
32 
32 
32 
3i 
S2 
32 
3i 
S2 
32 
3i 
32 
3i 
S2 
32 
3i 
32 
31 
32 
3i 
32 
3i 
32 
3i 
32 
3i 
3i 
32 
3i 
32 
3i 
3i 
32 
31 
3i 
32 
3i 
3i 
3i 
S2 

0.31182  2.0503 
0.31150   488 
0.31118   473 
0.31086   458 
0.31054   443 

60 
P 

1 

55 

54 
53 
52 
5i 
50 

49 
48 

47 
46 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

5 

6 

7 
8 

9 

43968  9.64313 

44020  9.64365 
046  9.64391 
072  9.64417 

89816  9.95335 
8°3  9-95329 
790  9-95323 
777  9-953I7 
764  9-95310 

48953  9-68978 

989  9.69010 
49026  9.69042 

062  9.69074 

098  9.69106 

0.31022  2.0428 
0.30990   413 
0.30958   398 
0.30926   383 
0.30894   368 

10 

ii 

12 
13 
14 

44098  9.64442 
124  9.64468 
151  9-64494 
177  9-6*5*9 
203  9.64545 

89752  9-95304 
739  9-95298 

713  9.95286 
700  9.95279 

49134  9.69138 

170  9.69170 
206  9.69202 

242  9.69234 
278  9.69266 

0.30862  2.0353 
0.30830   338 
0.30798   323 
0.30766   308 
0.30734   293 

15 

16 

17 
18 

19 

44229  9.64571 
255  9.64596 
281  9.64622 
307  9-64647 
333  9-64673 

89687  9.95273 

674  9-95267 
662  9.95261 
649  9-95254 
636  9.95248 

49315  9-69298 
351  9.69329 
387  9.69361 
423  9.69393 
459  9-69425 

0.30702  2.0278 
0.30671   263 
0.30639   248 
0.30607   233 
0.30575   2I9 

20 

21 
22 

23 

24 

44359  9-64698 
385  9.64724 
411  9.64749 

437  9.64775 
464  9.64800 

89623  9.95242 
610  9.95236 

597  9-95229 
584  9.95223 
571  9.95217 

49495  9-69457 
532  9.69488 
568  9.69520 
604  9.69552 
640  9.69584 

0.30543  2.0204 
0.30512   189 
0.30480   174 
0.30448   160 
0.30416   145 

25 

26 
27 
28 
29 

30 

3i 

32 
33 
34 

44490  9.64826 
516  9.64851 
542  9.64877 
568  9.64902 
594  9.64927 

89558  9-952II 
545  9-95204 
S32  9.95J98 
519  9.95192 
506  9-95185 

49677  9.69615 

713  9-69647 
749  9.69679 
786  9.69710 
822  9.69742 

0-30385  2.0130 

0.30353   JI5 
0.30321   101 
0.30290   086 
0.30258   072 

44620  9.64953 
646  9.64978 
672  9.65003 
698  9.65029 
724  9.65054 

89493  9-95I79 
480  9.95173 
467  9.95167 
454  9-95*60 
441  9.95154 

49858  9.69774 
894  9.69805 
931  9-69837 
967  9.69868 
50004  9.69900 

0.30226  2.0057 
0.30195   042 
0.30163   028 
0.30132   013 
0.30100  1.9999 

35 

36 
37 
38 
39 
40 

41 
42 

43 
44 
4T 

46 

47 
48 

49 

4475°  9-65079 
776  9.65104 
802  9.65130 
828  9.65155 
854  9-65180 

89428  9.95148 
415  9-95I4I 
402  9.95135 
389  9.95129 
376  9.95122 

50040  9.69932 
076  9.69963 
113  9.69995 
149  9.70026 
185  9.70058 

0.30068  1.9984 
0-30037   97° 
0.30005   955 
0.29974   941 
0.29942   926 

25 

24 
23 

22 
21 

44880  9.65205 
906  9.65230 
932  9.65255 
958  9.65281 
984  9.65306 

89363  9-95«6 
35°  9-95IIo 
337  9-95J03 
324  9.95097 
311  9.95090 

50222  9.70089 
258  9.70121 
295  9-70152 
331  9.70184 
368  9.70215 

0.29911  1.9912 
0.29879   897 
0.29848   883 
0.29816   868 
0.29785   854 

20 

19 
18 

3 

45010  9.65331 
036  9.65356 
062  9.65381 
088  9.65406 
114  9.65431 

89298  9.95084 
285  9.95078 
272  9-95071 
259  9-95065 
245  9.95059 

50404  9.70247 
441  9.70278 

477  9-70309 
514  9.70341 
550  9-70372 

0.29753  I-984° 
0.29722   825 
0.29691   811 
0.29659   797 
0.29628   782 

15 

14 
13 

12 
II 

50 

51 

52 
53 
54 

45140  9.65456 
166  9.65481 
192  9.65506 
218  9-6553I 
243  9-65556 

89232  9.95052 
219  9.95046 
206  9.95039 

193  9-95033 
180  9.95027 

50587  9.70404 
623  9.70435 
660  9.70466 
696  9.70498 
733  9-70529 

0.29596  1.9768 
0.29565   754 
0.29534   74° 
0.29502   725 
0.29471   711 

10 

9 
8 

I 

55 

56 
57 
58 

S 

45269  9.65580 
295  9-65605 
321  9.65630 
347  9.65655 
373  9-65680 
399  9.65705 

89167  9.95020 

*53  9-950I4 
140  9.95007 
127  9.95001 
114  9.94995 
101  9.94988 

50769  9.70560 
806  9.70592 
843  9.70623 
879  9.70654 
916  9.70685 
953  9-707*7 

0.29440  1.9697 
0.29408   683 
0.29377   669 
0.29346   654 
0.29315   640 
0.29283   626 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.  Log.Tan  Nat. 

f 

63 


27° 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat  Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

45399  9-65705 
425  9.65729 

45i  9-65754 
477  9-65779 
503  9.65804 

24 
25 
25 
25 
24 
25 
25 
24 
25 
25 
24 
25 
24 
25 
25 
24 
25 
24 

25 
24 
24 
25 
24 
25 
24 
24 
25 
24 
24 
25 
24 
24 
24 
24 
25 
24 
24 
24 

24 
24 
24 
25 
24 
24 
24 

24 
24 

24 
24 

23 
24 
24 
24 

24 
24 

24 
24 
23 
24 
24 

89101  9.94988 
087  9.94982 

074  9-94975 
061  9.94969 
048  9.94962 

6 

7 
6 

7 
6 

7 
6 

7 
6 

7 
6 
6 

7 
6 

7 
6 
7 
7 
6 

7 
6 

7 

6 

7 
6 

7 
6 

7 
7 
6 

7 
6 

7 
6 

7 
7 
6 

7 
6 

7 

I 

7 
7 
6 

7 
7 
6 

7 
7 
6 
7 
7 
6 

7 

7 
6 

7 
7 
7 

50953  9-70717 
989  9.70748 
51026  9.70779 

063  9.70810 

099  9.70841 

3i 
3i 
3i 
3i 
32 
3i 
3i 
31 
3i 
3i 
3i 
3i 
3i 
32 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
30 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
3i 
30 
3i 
3i 
3i 
3i 
30 
3i 
3i 
30 
31 
3i 
3i 
30 
31 
3i 
30 
3i 
30 
3i 
3i 
30 
3i 
30 
3i 
30 
3i 
3° 
31 
30 

0.29283  1.9626 
0.29252   612 
0.29221   598 
0.29190   584 
0.29159   570 

60 

If 
i 

5 

6 

7 

8 

9 

45529  9-65828 
554  9-65853 
580  9.65878 
606  9.65902 
632  9-65927 

89035  9-94956 
021  9.94949 
008  9.94943 
88995  9.94936 
981  9.94930 

51136  9-70873 

173  9.70904 
209  9.70935 
246  9.70966 

283  9.70997 

0.29127  1.9556 
0.29096   542 
0.29065   528 
0.29034   514 
0.29003   500 

55 

54 
53 
52 
5i 
50 

49 
48 

4£ 
46 

10 

ii 

12 
13 
14 

45658  9.65952 
684  9.65976 
710  9.66001 
736  9.66025 
762  9.66050 

88968  9.94923 
955  9-949I7 
942  9-94911 
928  9.94904 
915  9.94898 

51319  9.71028 

356  9.71059 

393  9-71090 
430  9.71121 
467  9.71153 

0.28972  1.9486 
0.28941   472 
0.28910   458 
0.28879   444 
0.28847   430 

15 

16 

17 

18 

19 
20 

21 
22 

23 
24 

2T 

26 

3 

29 
30 

3i 

32 
33 
34 

45787  9.66075 
813  9.66099 
839  9.66124 
865  9.66148 
891  9.66173 

88902  9.94891 
888  9.94885 
875  9-94878 
862  9.94871 
848  9.94865 

51503  9.71184 
540  9.71215 
577  9.71246 
614  9.71277 
651  9.71308 

0.28816  1.9416 
0.28785   402 
0.28754   388 
0.28723   375 
0.28692   361 

45 

44 
43 
42 
41 

45917  9.66197 
942  9.66221 
968  9.66246 
994  9.66270 
46020  9.66295 

88835  9-94858 
822  9.94852 
808  9.94845 

795  9-94839 
782  9.94832 

51688  9.71339 
724  9.71370 
761  9.71401 
798  9.7143* 
835  9.71462 

0.28661  1.9347 
0.28630   333 
0.28599   319 
0.28569   306 
0.28538   292 

40 

P 
1 

46046  9.66319 
072  9.66343 
097  9.66368 
123  9.66392 
149  9.66416 

88768  9.94826 
755  9.94819 
741  9.94813 
728  9.94806 
715  9-94799 

51872  9.71493 
909  9-7I524 
946  9.71555 
983  9-7I586 
52020  9.71617 

0.28507  1.9278 
0.28476   265 
0.28445   251 
0.28414   237 
0.28383   223 

35 

34 
33 
32 
3i 

46175  9.66441 

201  9.66465 
226  9.66489 
252  9.66513 
278  9.66537 

88701  9-94793 
688  9.94786 
674  9-94780 
661  9.94773 
647  9-94767 

52057  9.71648 
094  9.71679 
131  9.71709 
168  9.71740 
205  9.71771 

0.28352  1.9210 
0.28321   196 
0.28291   183 
0.28260   169 
0.28229   155 

30 

29 
28 
27 
26 

35 

36 
37 
38 
39 

46304  9.66562 
330  9.66586 

355  9.66610 
381  9.66634 
407  9.66658 

88634  9-94760 
620  9-94753 
607  9-94747 
593  9-94740 
580  9-94734 

52242  9.71802 

279  9-7I833 
316  9.71863 
353  9-71894 
390  971925 

0.28198  1.9142 
0.28167   128 
0.28137   115 
0.28106   101 
0.28075   088 

25 

24 
23 

22 
21 

20 

19 
18 

% 

40 

41 
42 
43 
44 
45" 
46 

47 
48 

49 
50 

5i 

52 
53 

54 

46433  9.66682 
458  9.66706 
484  9.66731 
510  9.66755 
536  9.66779 

88566  9.94727 
553  9-94720 
539  9-947T4 
526  9.94707 
512  9.94700 

52427  9.71955 
464  9.71986 
501  9.72017 
538  9.72048 
575  9-72078 

0.28045  1.9074 
0.28014   061 
0.27983   047 
0.27952   034 

0.27922    020 

46561  9.66803 
587  9.66827 
613  9.66851 
639  9.66875 
664  9.66899 

88499  9-94694 
485  9.94687 
472  9.94680 
458  9.94674 
445  9-94667 

52613  9.72109 
650  9.72140 
687  9.72170 
724  9.72201 
761  9.72231 

0.27891  1.9007 
0.27860  1.8993 
0.27830    980 
0.27799    967 

0.27769   953 

15 

14 
13 

12 
II 

46690  9.66922 
716  9.66946 
742  9.66970 
767  9.66994 
793  9.67018 

88431  9.94660 
417  9.94654 
404  9.94647 
390  9.94640 
377  9.94634 

52798  9.72262 
836  9.72293 

873  972323 
910  9.72354 

947  9-72384 

0.27738  1.8940 
0.27707   927 
0.27677   913 
0.27646   900 
0.27616   887 

10 

9 
8 

i 

5 

4 
3 

2 

I 

0 

55 

56 

i 
18 

46819  9.67042 
844  9.67066 
870  9.67090 
896  9.67113 
921  9.67137 
947  9.67161 

88363  9.94627 
349  9.94620 
336  9.94614 
322  9.94607 
308  9.94600 
295  9-94593 

52985  9.7241$ 
53022  9.72445 
059  9.72476 
096  9.72506 
134  9.72537 
171  9.72567 

0.27585  1.8873 

0-27555   86° 
0.27524   847 
0.27494   834 
0.27463   820 
0.27433   807 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.Tan  Nat. 

/ 

62C 


28C 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d.  |Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

46947  9.67161 
973  9.67185 
999  9-67208 
47024  9.67232 
050  9.67256 

24 

23 
24 
24 
24 
23 
24 
23 
24 
24 
23 
24 
23 
24 
23 
24 

23 
24 

23 
24 
23 
24 
23 
23 
24 

23 
23 
24 
23 
23 
24 
23 
23 
23 
23 
24 
23 
23 
23 
23 
23 
23 
23 
23 
23 
24 
23 
23 

22 
23 
23 
23 
23 
23 
23 
23 
23 
23 
22 

23 

88295  9-94593 
281  9.94587 
267  9-9458o 
254  9-94573 
240  9.94567 

6 
7 
7 
6 

7 
7 

7 
6 

7 
7 

7 
6 

7 
7 
7 

7 
6 

7 
7 
7 

7 
6 

7 
7 
7 
7 
7 
6 

7 
7 
7 
7 
7 
7 
7 
6 
7 
7 
7 
7 
7 
7 
7 
7 
7 

7 
6 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

53171  9.72567 
208  9.72598 
246  9.72628 
283  9.72659 
320  9.72689 

3i 
30 
3i 
30 
3i 
30 
30 
3i 
3° 
3i 
30 
3° 
3i 
30 
30 

11 

% 

3i 
3° 
30 
30 
3° 
3i 
30 
3° 
3° 
3° 
30 
3i 
3° 
30 
30 
30 
30 
30 
30 
30 
30 
30 
3° 
30 
30 
3° 
30 
3° 
30 
30 
3° 
30 
30 
29 
30 
30 
30 
30 
30 
29 
30 

0.27433  1.8807 
0.27402   794 
0.27372   781 
0.27341   768 
0.27311   755 

60 

1 

5 

6 

7 
8 

9 

To 

ii 

12 
13 
14 

47076  9.67280 
10  1  9.67303 
127  9.67327 

153  9-67350 
178  9.67374 

88226  9.94560 
213  9-94553 
*99  9-94546 
J85  9-94540 
172  9.94533 

53358  9-72720 
395  9-72750 
432  9.72780 
470  9.72811 
507  9.72841 

0.27280  1.8741 
0.27250   728 
0.27220   715 
0.27189   702 
0.27159   689 

55 

54 
53 
52 
5i 

47204  9.67398 
229  9.67421 
255  9.67445 
281  9.674613 
306  9-67492 

88158  9.94526 
144  9.94519 
13°  9-945I3 
IJ7  9-94506 
103  9.94499 

53545  9-72872 
582  9.72902 
620  9.72932 
657  9.72963 
694  9.72993 

0.27128  1.8676 
0.27098   663 
0.27068   650 
0.27037   637 
0.27007   624 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

15 

16 

3 

19 

47332  9.67515 
358  9-67539 
383  9.67562 
409  9.67586 
434  9-67609 

88089  9.94492 

075  9-94485 
062  9.94479 
048  9.94472 
034  9.94465 

53732  9-73023 
769  9.73054 
807  9.83084 
844  9.73114 
882  9.73144 

0.26977  1.8611 
0.26946   598 
0.26916   585 
0.26886   572 
0.26856   559 

20 

21 
22 

23 
24 

47460  9.67633 
486  9.67656 
511  9.67680 

537  9.67703 
562  9.67726 

88020  9.94458 
006  9.94451 
87993  9-94445 
979  9-94438 
965  9.94431 

53920  9.73175 
957  9-7320$ 
995  9-73235 
54032  9.73265 

070  9.73295 

0.26825  1.8546 
0.26795   533 
0.26765   520 
0.26735   507 
0.26705   495 

40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 

26 
27 
28 
29 
30 
3i 
32 
33 
34 

47588  9.67750 
614  9.67773 
639  9.67796 
665  9.67820 
690  9.67843 

87951  9.94424 
937  9-944I7 
923  9-94410 
909  9.94404 
896  9-94397 

54107  9.73326 
145  9.73356 
183  9.73386 

220  9.73416 
258  9.73446 

0.26674  1.8482 
0.26644   469 
0.26614   456 
0.26584   443 
0.26554   430 

47716  9.67866 
741  9.67890 
767  9.67913 
793  9-67936 
818  9.67959 

87882  9.94390 
868  9.94383 
854  9-94376 
840  9.94369 
826  9.94362 

54296  9-73476 

333  9-73507 

409  9-73567 
446  9-73597 

0.26524  1.8418 
0.26493   405 
0.26463   392 
0.26433   379 
0.26403   367 

35 

36 
37 
38 
39 

47844  9.67982 
869  9.68006 
895  9.68029 
920  9.68052 
946  9.68075 

87812  9.94355 
798  9-94349 
784  9-94342 
770  9-94335 
756  9.94328 

54484  9.73627 
522  9.73657 
560  9.73687 
597  9-737I7 
635  9-73747 

0.26373  1.8354 
0.26343   341 
0.26313   329 
0.26283   316 
0.26253   303 

25 

24 
23 

22 
21 

20 

19 
18 

3 

40 

41 
42 

43 
44 

47971  9.68098 
997  9.68121 
48022  9.68144 
048  9.68167 
073  9.68190 

87743  9-94321 
729  9.94314 
715  9.94307 
701  9.94300 
687  9.94293 

54673  9-73777 
711  9.73807 
748  9.73837 
786  9.73867 
824  9.73897 

0.26223  1.8291 
0.26193   278 
0.26163   265 
0.26133   253 
0.26103   240 

45 

46 

47 
48 

49 

48099  9.68213 
124  9.68237 
ISO  9.68260 
175  9.68283 

201  9.68305 

87673  9.94286 
659  9.94279 
645  9.94273 
631  9.94266 
617  9.94259 

54862  9.73927 
9°o  9-73957 
938  9-73987 
975  9-740I7 
55oi3  9-74047 

0.26073  1.8228 
0.26043   215 

O.260I3    202 
0.25983    190 
0.25953    177 

15 

14 
13 

12 
II 

50 

Si 
52 
53 
54 
5T 
56 

i 

J9o 

48226  9.68328 
252  9.68351 
277  9.68374 
303  9.68397 
328  9.68420 

87603  9.94252 
589  9.94245 
575  9-94238 
561  9.94231 
546  9.94224 

55051  9.74077 
089  9.74107 
127  9.74137 
165  9.74166 
203  9.74196 

0.25923  I.8l65 
0.25893     152 
0.25863    140 
0.25834     127 
0.25804     115 

10 

9 

8 

7 
6 

48354  9.68443 

379  9.68466 
405  9.68489 
430  9.68512 
456  9.68534 
481  9.68557 

87532  9.94217 
518  9.94210 
504  9.94203 
490  9.94196 
476  9.94189 
462  9.94182 

55241  9.74226 
279  9-74256 
317  9.74286 
355  9-743I6 
393  9-74345 
43i  9-74375 

0.25774  I.8I03 
0.25744    090 
0.25714    078 
0.25684    065 
0-25655    053 
0.25625    040 

5 

4 
3 

2 

I 

0 

Nat.CoSLog.  d. 

Nat.  Sin  Log.  d. 

Nat.CotLog.|c.d. 

Log.Tan  Nat. 

f 

61 


29° 


/ 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

-y 

6 

7 
8 

9 

48481  9.68557 
506  9.68580 
532  9-68603 
557  9.68625 
583  9.68648 

23 
23 

22 
23 
23 

23 
22 

23 
23 
22 

23 
22 

23 
23 
22 

23 
22 

23 
22 

23 
22 

23 
22 

23 

22 
22 

23 

22 

23 

22 
22 

23 
22 
22 
22 

23 
22 

22 
22 
22 

23 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

87462  9.94182 
448  9.94175 
434  9.94168 
420  9.94161 
406  9.94I54 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
8 

7 
7 
7 
7 
7 
7 
7 
7 
7 

I 

7 
7 
7 
7 
7 
7 
7 
8 

7 
7 
7 

7 
7 
8 

7 
7 
7 
7 
8 

7 
7 

7 
7 
7 

7 
7 
8 

7 
7 
7 
8 

7 

55431  9.74375 
409  9.74405 

507  9.74435 

545  9-74465 
583  9.74494 

30 
30 
30 
29 

30 
30 
29 
30 
30 
30 
29 
3° 
30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 

29 
29 
30 
29 
29 
29 
30 
29 
29 
29 
30 
29 
29 

0.25625  1.8040 
0.25595  028 
0.25565  016 

0-25535  003 
0.25506  1.7991 

60 

ii 

% 

48608  9.68671 
634  9.68694 
6^9  9.68716 
684  9-68739 
710  9.68762 

87391  9.94147 
377  9.94140 
363  9.94133 
349  9.94126 
335  9.94II9 

55621  9.74524 
659  9-74554 
697  9-74583 
736  9.74613 
774  9.74643 

0.25476  1.7979 
0.25446  966 
0.25417  954 
0.25387  942 
0.25357  930 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 

14 

48735  9.68784 
761  9.68807 
786  9.68829 
811  9.68852 
837  9.68875 

87321  9.94112 
306  9.94105 
292  9.94098 
278  9.94090 
264  9.94083 

55812  9.74673 
850  9.74702 
888  9.74732 
926  9.74762 
964  9.74791 

0.25327  1.7917 
0.25298  905 
0.25268  893 
0.25238  88  1 
0.25209  868 

50 

49 
48 
47 
46 

45 

44 
43 
42 
4i 
40 

P 
1 

15 

16 

17 
18 

19 
20 

21 
22 

23 

24 

25 

26 

29 

48862  9.68897 
888  9.68920 
913  9.68942 
938  9-68965 
964  9.68987 

87250  9.94076 
235  9-94069 

221  9.94062 
207  9.94055 
193  9.94048 

56003  9.74821 
041  9.74851 
079  9.74880 
117  9.74910 
156  9.74939 

0.25179  1.7856 
0.25149  844 
0.25120  832 
0.25090  820 
0.25061  808 

48989  9.69010 
49014  9.69032 
040  9.69055 
065  9.69077 
090  9.69100 

87178  9.94041 
I64  9.94034 
150  9.94027 
136  9.94020 
121  9.94012 

56194  9.74969 
232  9.74998 
270  9.75028 
309  9.75058 
347  9.75087 

0.25031  1.7796 
0.25002  783 
0.24972  771 
0.24942  759 
0.24913  747 

49116  9.69122 
141  9.69144 
166  9.69167 
192  9.69189 
217  9.69212 

87107  9.94005 
093  9.93998 
079  9-93991 
064  9.93984 

050  9-93977 

56385  9.75H7 
424  9-75I46 
462  9.75176 
501  9.75205 
539  9.75235 

0.24883  1.7735 
0.24854  723 
0.24824  711 
0.2479^  699 
0.24765  687 

35 

34 
33 
32 
3i 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 
45 

46 

47 

48 

49 

49242  9.69234 
268  9.69256 

293  9.69279 
318  9.69301 
344  9.69323 

87036  9.93970 
021  9.93963 
007  9-93955 
86993  9.93948 
978  9.93941 

56577  9-75264 
616  9.75294 
654  9.75323 
693  9-75353 
73i  975382 

0.24736  1.7675 
0.24706  663 
0.24677  651 
0.24647  639 
0.24618  627 

30 

29 
28 
27 
26 

49369  9.69345 
394  9.69368 
419  9.69390 
445  9-694I2 
470  9.69434 

86964  9.93934 
949  9.93927 
935  9-93920 
921  9.93912 
906  9.93905 

56769  9.75411 
808  9.75441 
846  9.75470 
885  9-75500 
923  975529 

0.24589  1.7615 
0.24559  603 
0.24530  59i 
0.24500  579 
0.24471  567 

25 

24 
23 

22 
21 

49495  9-69456 
521  9.69479 
546  9.69501 
571  9-69523 
596  9.69545 

86892  9.93898 
878  9.93891 
863  9-93884 
849  9.93876 
834  9.93869 

56962  9.75558 
57000  9.75588 
°39  9.75617 
078  9.75647 
116  9*75676 

0.24442  1.7556 
0.24412  544 
0.24383  532 
0-24353  520 
0.24324  508 

20 

19 
18 

17 
16 

15 
14 
13 

12 
II 

10 
1 
| 

49622  9.69567 

647  9-69589 
672  9.69611 
697  9.69633 
723  9.69655 

86820  9.93862 
805  9.93855 
791  9.93847 
777  9.93840 
762  9.93833 

57155  975705 
X93  9-75735 
232  9-75764 
271  9-75793 
309  9.75822 

0.24295  1.7496 
0.24265  485 
0.24236  473 
0.24207  461 
0.24178  449 

50 

5i 

52 
53 
54 

49748  9.69677 

773  9-69699 
798  9.69721 
824  9.69743 
849  9.69765 

86748  9.93826 

733  9-938I9 
719  9.93811 
704  9.93804 
690  9-93797 

57348  9.75852 
386  9.75881 
425  9-759io 
464  9-75939 
503  9.75969 

0.24148  1.7437 
0.24119  426 
0*24090  414 
0.24061  402 
0.24031  391 

55 

56 
57 
58 

fo 

49874  9.69787 

924  9.69831 
950  9.69853 
975  9.69875 
50000  9-69897 

86675  9.93789 
66  1  9.93782 
646  9-93775 
632  9-93768 
617  9-9376o 
603  9-93753 

57541  975998 
580  9.76027 
619  9.76056 
657  9.76086 
696  9.76115 
735  9-76I44 

0.24002  1.7379 
0.23973  367 
0-23944  355 
0.23914  344 
0.23885  332 
0.23856  321 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.Log.TanNat.|  ' 

60C 


30° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLogJc.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

~y 

6 

1 

9 

50000  9.69897 
025  9.69919 
050  9.69941 
076  9.69963 
101  9.69984 

22 
22 
22 
21 
22 
22 
22 
22 
21 
22 
22 
22 
21 
22 
22 
21 
22 
21 
22 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
21 
22 
21 
21 
22 
21 
21 
21 
22 
21 
21 
21 
22 
21 
21 
21 
21 
21 
22 
21 
21 
21 
21 
21 
21 

86603  9-93753 
588  9.93746 
573  9-93738 
559  9-93731 
544  9-93724 

7 
8 

7 

7 
7 
8 
7 

7 
7 
8 

7 
8 

7 
7 
8 

7 
7 
8 

7 
8 

7 
7 
8 

I 

7 
8 

7 
7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

8 
I 

7 
8 

7 

57735  9-76I44 
774  9.76173 
813  9.76202 
851  9.76231 
890  9.76261 

29 
29 
29 
30 
29 

29 
29 
29 
29 

29 
29 
29 

29 
29 

29 
29 
3° 

29 
29 
28 
29 
29 
29 
29 

29 
29 
29 

29 

29 
29 

29 
29 
28 
29 
29 
29 
29 
29 
28 
29 

29 
29 

29 
28 
29 

29 
29 

28 
29 

29 
28 
29 
29 
29 
28 
29 
28 
29 

29 
28 

0.23856  1.7321 
0.23827   309 
0.23798   297 
0.23769   286 
0-23739   274 

60 

59 
58 
57 
56 

50126  9.70006 
151  9.70028 
176  9.70050 

201  9.70072 
227  9.70093 

86530  9-937I7 
515  9-93709 
501  9.93702 

486  9-93695 
471  9.93607 

57929  9-76290 
968  9.76319 
58007  9.76348 
046  9-76377 
085  9.76406 

0.23710  1.7262 
0.23681   251 
0.23652   239 
0.23623   228 
0.23594   216 

55 

54 
53 
52 
5J 

10 

ii 

12 

J3 

14 

50252  9.70II5 
277  9.70137 
302  9.70159 
327  9.70180 
352  9.70202 

86457  9.93680 

442  9-9367? 
427  9.93665 
413  9-93658 
398  9.93650 

58124  9.76435 
162  9.76464 

201  9.76493 
240  9.76522 
279  9-7655I 

0.23565  1.7205 
0.23530   193 
0.23507   182 
0.23478   170 
0.23449  .  159 

50 

49 
48 
47 
46 

15 

16 

3 

19 

50377  9.70224 
403  9.76245 

453  9.70288 
478  9-703IO 

86384  9.93643 
369  9.93636 
354  9-93628 
340  9.93621 
325  9-936I4 

58318  9.76580 

357  9.76609 
396  9.76639 
435  9-76668 
474  9.76697 

0.23420  1.7147 
0.23391   136 
0.23361   124 
0.23332   113 

0.23303    102 

45 

44 
43 
42 
41 

20 

21 
22 

23 

24 

~25 

26 
27 
28 
29 

50503  9-70332 
528  9.70353 
553  9-70375 
578  9-70396 
603  9.70418 

86310  9.93606 
295  9-93599 
281  9-93591 
266  9.93584 
251  9-93577 

58513  9-76725 
552  9-76754 
591  9.76783 
631  9.76812 
670  9.76841 

0.232-55  1.7090 
0.23246    079 
0.23217    067 
0.23188    056 
0.23159    045 

40 

39 
38 

| 

50628  9.70439 
654  9.70461 
679  9.70482 
704  9.70504 
729  9.70525 

86237  9-93569 

222  9.93562 

207  9-93554 
!92  9-93547 
J78  9-93539 

58709  9.76870 
748  9-76899 
787  9.76928 
826  9.76957 
865  9.76986 

0.23130  1.7033 
0.23IOI     022 

0.23072   on 

0.23043  1.6999 
0.23014    988 

35 

34 
33 
32 
3i 

30 

31 
32 
33 
34 

50754  9.70547 
779  9-70568 
804  9.70590 
829  9.70611 
854  9.70633 

86163  9-93532 
148  9.93525 
i33  9-935I7 
"9  9-935Io 
104  9-93502 

58905  9.77015 

944  9-77044 
983  9.77073 
59022  9.77101 
061  9.77130 

0.22985  1.6977 
0.22956     9b5 

0.22927   954 
0.22899   943 
0.22870   932 

30 

29 
28 

Z 

35 

36 
37 
38 
39 

50879  9.70654 
904  9.70675 
929  9.70697 
954  9-707I8 
979  9-70739 

86089  9.93495 
074  9.93487 
°59  9-9348o 
045  9-93472 
°3°  9-93465 

59101  9.77159 
140  9.77188 
179  9.77217 
218  977246 
258  9-77274 

0.22841  1.6920 
0.22812   909 
0.22783   898 
0.22754   887 
0.22726   875 

25 

24 
23 

22 
21 

40 

41 

42 

43 
44 
4T 

46 

? 

49 

51004  9.70761 
029  9.70782 
054  9.70803 
079  9.70824 
104  9.70846 

86015  9-93457 
°°o  9-9345° 
85985  9-93442 
970  9-93435 
956  9-93427 

59297  9.77303 
336  9-77332 
376  9-7736i 
415  9-77390 
454  9.77418 

0.22697  1.6864 
0.22668   853 
0.22639   842 
0.22610   831 
0.22582   820 

20 

IQ 

17 

16 

51129  9.70^67 
154  9.70888 
179  9.70909 
204  9.70931 
229  9.709^2 

85941  9.93420 
926  9.93412 
911  9.93405 
896  9-93397 
88  1  9-93390 

59494  9-77447 
533  9-77476 
573  9-77505 
612  9.77533 
651  9.77562 

0.22553  i.  6808 
0.22524   797 
0.22495   786 
0.22467   775 
0.22438   764 

15 

14 
13 

12 
II 

To^ 

9 
8 

7 
6 

50 

5i 

52 
53 
54 

51254  9.70973 
279  9.70994 
304  9.71015 
329  9.71036 
354  9-7!058 

85866  9.93382 
851  9-93375 
836  9-93367 
821  9.93360 
806  9.93352 

59691  9.77591 
730  9.77619 
770  9.77648 
809  9.77677 
849  9.77706 

0.22409  1.6753 
0.22381   742 
0.22352   731 
0.22323   720 
0.22294   709 

55 

56 

i 

lo 

5J379  9-7xo79 
404  9.71100 
429  9.71121 
454  9.71142 
479  9.71163 
504  9.71184 

85792  9.93344 
777  9-93337 
762  9.93329 
747  9-93322 
732  9.93314 
717  9-93307 

59888  9.77734 
928  9.77763 
967  9.77791 
60007  9.77820 
046  9.77849 
086  9.77877 

0.22266  1.6698 
0.22237   687 
0.22209   676 
0.22180   665 
0.22151   654 
0.22123   643 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  CotLog.|c.d.  Log.  Tan  Nat. 

; 

59 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d.  |Nat.TanLog.  c.d.|Log.Cot  Nat.| 

0 

I 

2 

3 
4 

,51504  9.71184 
529  9.71205 
554  9.71226 
579  9.71247 
604  9.71268 

21 
21 
21 
21 
21 
21 
21 
21 
21 
2O 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
2O 
21 
21 
21 
2O 
21 
21 
20 
21 
21 
20 
21 
20 
21 
20 
21 
20 
21 
21 
20 
20 
21 
20 
21 
2O 
21 
2O 
2O 
21 
2O 

£ 

20 
20 

21 
20 
20 
21 
2O 
20 

85717  9-93307 
702  9.93299 
687  9.93291 
672  9.93284 
657  9-9327<5 

8 
8 

7 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 

7 
8 
8 
8 

7 

8 
8 
8 
7 
8 
8 
8 
8 
7 
8 
8 
8 

7 
8 

8 
8 
8 
8 
7 
8 
8 
8 
8 
8 
8 

7 
8 
8 
8 
8 

60086  9.77877 
126  9.77906 
165  9-77935 
205  9-77963 
245  9-77992 

29 

29 
28 
29 
28 

29 
28 
29 
29 
28 
29 
28 

29 
28 
29 
28 

29 
28 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
28 

29 
28 
28 
28 

29 
28 
28 
28 
28 

28 
28 
28 
28 
29 
28 
28 
28 

0.22123  1.6643 
0.22094  632 
0.22065  621 
0.22037  6  10 
0.22008  599 

60 

59 
58 
57 
56 

5 

6 

7 
8 

9 
10 

ii 

12 
13 
14 

51628  9.71289 
653  9-7i3Jo 
678  9.71331 
703  9-7I352 
728  9.71373 

85642  9.93269 
627  9.93261 
612  9.93253 
597  9-93246 
582  9.93238 

60284  9.78020 
324  9.78049 
364  9.78077 
403  9.78106 
443  9-78I35 

0.21980  1.6588 
0.21951  577 
0.21923  566 
0.21894  555 
0.21865  545 

55 

54 
53 
52 
5i 

51753  9-7I393 
778  9.71414 
803  9.71435 
828  9.71456 
852  9.71477 

85567  9-93230 
551  9.93223 
536  9-932I5 
521  9-93207 
506  9.93200 

60483  9.78163 
522  9.78192 
562  9.78220 
602  9.78249 
642  9.78277 

0.21837  1.6534 
0.21808  523 
0.21780  512 
0.21751  501 
0.21723  490 

50 

49 
48 

47 
46 
45 

44 
43 
42 
41 

15 

16 

17 
18 
19 

51877  9.71498 
902  9.71519 
927  9.71539 
952  9-7i56o 
977  9.71581 

85491  9.93192 
476  9.93184 
461  9.93177 
446  9-93I69 
431  9.93161 

60681  9.78306 
721  9.78334 
761  9.78363 
801  9.78391 
841  9.78419 

0.21694  1.6479 
0.21666  469 
0.21637  458 
0.21609  447 
0.21581  436 

20 

21 
22 
23 
24 

52002  9.71602 
026  9.71622 
051  9.71643 
076  9.71664 
101  9.71685 

85416  9.93154 
401  9.93146 
385  9-93I38 
370  9.93131 
355  9-93I23 

60881  9.78448 
921  9.78476 
960  9.78505 
61000  9.78533 
040  9.78562 

0.21552  1.6426 
0.21524  415 
0.21495  404 
0.21467  393 
0.21438  383 

40 

It 
1 

25 

26 

27 
28 
29 

52126  9.71705 
151  9.71726 
J75  9-7I747 

200  9.71767 
225  9.71788 

85340  9-93II5 
325  9-93io8 
310  9.93100 
294  9-93092 
279  9.93084 

01080  9.78590 
120  9.78618 
160  9.78647 

200  9.78675 
240  9.78704 

0.21410  1.6372 
0.21382  361 
0.21353  35i 
0.21325  340 
0.21296  329 

35 

34 
33 
32 
3i 

30 

3i 
32 
33 
34 

52250  9.71809 
275  9.71829 
299  9.71850 
324  9.71870 

349  9.71891 

85264  9.93077 
249  9.93069 
234  9-9306I 
218  9.93053 
203  9.93046 

61280  9.78732 
320  9.78760 
360  9.78789 
400  9.78817 
440  9.78845 

0.21268  1.6319 
0.21240  308 

0.2I2II  297 
O.2II83  287 
0.2II55  276 

30 

29 
28 

27 
26 

35 

36 

P 

39 

52374  9.71911 
399  9-7I932 
423  9-7I952 
448  9.71973 
473  9.71994 

85188  9.93038 
173  9-93030 
157  9-93022 
142  9.93014 
127  9.93007 

61480  9.78874 
520  9.78902 
56l  9.78930 

601  9.78959 
641  9.78987 

O.2II26  1.6265 
0.21098  255 
O.2I070  244 
0.2I04I  234 
O.2IOI3  223 

25 

24 
23 

22 
21 

40 

41 
42 

43 
44 
45 

46 

3 

49 

52498  9.72014 
522  9.72034 
547  9.72055 
572  9.72075 
597  9.72096 

85112  9.92999 
096  9.92991 
08  i  9.92983 
066  9.92976 
051  9.92968 

61681  9.79015 
721  9.79043 
761  9.79072 
801  9.79100 
842  9.79128 

O.2098g  I.62I2 
0.20957  202 
0.2O928  191 

0.20900  181 

0.20872  170 

20 

19 
18 

17 
16 

52621  9.72116 
646  9.72137 
671  9.72157 
696  9.72177 
720  9.72198 

85035  9-92960 

020  9.92952 
005  9.92944 
84989  9.92936 

974  9.92929 

61882  9.79156 
922  9.79185 
962  9.79213 
62003  9-7924J 
043  9.79269 

0.20844  1.6160 
0.20815  149 
0.20787  139 
0.20759  128 
0.20731  118 

15 

14 
13 

12 
II 

50 

51 

52 
53 
54 

52745  9.72218 
770  9.72238 
794  9.72259 
819  9.72279 
844  9.72299 

84959  9-92921 
943  9-929I3 

913  9.92897 
897  9.92889 

62083  9.79297 
124  9.79326 
I64  9-79354 
204  9-79382 
245  9-79410 

0.20703  1.6107 
0.20674  097 
0.20646  087 
0.20618  076 
0.20590  066 

10 

9 
8 

7 
6 

55 

56 
57 
58 

§9o 

52869  9.72320 
893  9.72340 
918  9.72360 
943  9.72381 
967  9.72401 
992  9.72421 

84882  9.92881 
866  9.92874 
851  9.92866 
836  9.92858 
820  9.92850 
805  9.92842 

62285  9-79438 
325  9-79466 
366  9.79495 
406  9.79523 
446  9-79551 
487  979579 

0.20562  1.6055 
0.20534  045 
0.20505  034 
0.20477  024 
0.20449  014 
0.20421  003 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d. 

Log.TanNat. 

f 

58° 


32C 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d 

Log.  Cot  Nat. 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 
48 

47 
46 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

0 

I 

2 

3 
4 
5 

6 

9 

52992  9.72421 
53017  9.72441 
041  9.72461 
066  9.72482 
091  9.72502 

20 
20 

21 
2O 
20 
20 
20 
20 
20 
2O 
21 
2O 
2O 
20 
20 
2O 
2O 
2O 
2O 
20 
20 
20 
2O 
19 
2O 
20 
20 
20 
20 
2O 

19 
2O 
2O 
20 
20 

19 
2O 
2O 
2O 

19 
2O 
2O 

19 
2O 

20 

19 
2O 
2O 
19 
20 

19 
2O 

19 
2O 

19 

20 

19 
2O 

19 
2O 

84805  9.92842 
789  9.92834 
774  9.92826 
759  9.92818 
743  9.92810 

8 
8 
8 
8 

7 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 

9 
8 
8 
8 
8 
8 
8 
8 
8 

9 

8 
8 
8 
8 
8 
8 

9 
8 

8 
8 
8 
8 

62487  9.79579 
527  9.79607 
568  9.79635 
608  9.79663 
649  9.79691 

28 
28 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 

27 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
28 

27 
28 
28 
27 
28 
28 

27 
28 
28 

0.20421  1.6003 
0.20393  1.5993 
0.20365   983 
0-20337   972 
0.20309   962 

53115  9.72522 
140  9.72542 
164  9.72562 
189  9.72582 
214  9.72602 

84728  9.92803 
712  9.92795 
697  9-92787 
68  1  9.92779 
666  9.92771 

62689  9.79719 
730  9.79747 
770  9.79776 
8  ii  9.79804 
852  9-79832 

0.20281  1.5952 
0.20253   941 
0.20224   931 
0.20196   921 
0.20168   911 

10 

ii 

12 
13 
14 

53238  9.72622 
263  9.72643 
288  9.72663 
312  9-72683 
337  9-72703 

84650  9.92763 

635  9-92755 
619  9.92747 

604  9-92739 
588  9.92731 

62892  9.79860 
933  9-79888 
973  9-799I6 
63014  9.79944 
055  9-79972 

0.20140  1.5900 

0.20II2    890 
0.20084    880 
0.20056    869 
0.2O028    859 

15 

16 

17 

18 

19 

53361  9.72723 
386  9.72743 
411  9.72763 

435  972783 
460  9.72803 

84573  9-92723 
557  9.92715 
542  9.92707 
526  9.92699 
511  9.92691 

63095  9.80000 
136  9.80028 
177  9.80056 
217  9.80084 
258  9.80112 

O.20OOO  1.5849 
0.19972    839 
0.19944    829 
0.19916   818 
0.19888   808 

20 

21 
22 

23 

24 

53484  9.72823 
509  9.72843 
534  9.72863 
558  9.72883 
583  9.72902 

84495  9.92683 
480  9*92675 
464  9.92667 
448  9.92659 

63299  9.80140 
340  9.80168 
380  9.80195 
421  9.80223 
462  9.80251 

0.19860  1.5798 
0.19832   788 
0.19805   778 
0.19777   768 
0.19749   757 

25 

26 
27 
28 
29 

30 

3i 
32 
33 
34 
35 
36 

3 

39 
40 

41 
42 
43 

44 

53607  9.72922 
632  9.72942 
656  9.72962 
68  1  9.72982 
70S  9-73002 

84417  9.92643 
402  9.92635 
386  9.92627 
370  9.92619 
355  9.92611 
84339  9-92603 
324  9-92595 
308  9.92587 
292  9.92579 
277  9-92571 

63503  9-80279 
544  9.80307 
584  9.80335 
625  9.80363 
666  9.80391 

0.19721  1.5747 
0.19693   737 
0.19665   727 
0.19637   717 
0.19609   707 

5373°  9-73022 
754  9-7304I 
779  9-7306I 
804  9.73081 
828  9.73101 

63707  9.80419 
748  9.80447 
789  9.80474 
830  9.80502 
871  9.80530 

0.19581  1.5697 
0.19553   687 
0.19526   677 
0.19498   667 
0.19470   657 

53853  9-73J2I 
877  9-73T40 
902  9.73160 
926  9.73180 
951  9.73200 

84261  9.92563 

245  9-92555 
230  9.92546 
214  9.92538 
198  9.92530 

63912  9.80558 
953  9-80586 
994  9.80614 
64035  9.80642 
076  9.80669 

0.19442  1.5647 
0.19414   637 
0.19386   627 
0.19358   617 
0.19331   607 

25 

24 
23 

22 
21 

20 

19 
18 

17 
16 

53975  9.73219 
54000  9.73239 
024  9.73259 
049  9.73278 
073  9-73298 

84182  9.92522 
167  9.92514 
151  9.92506 
135  9-92498 

120  9.92490 

64117  9.80697 
158  9.80725 
199  9.80753 
240  9.80781 
281  9.80808 

0.19303  1.5597 
0.19275   587 
0.19247   577 
0.19219   567 
0.19192   557 

45 

46 

47 
48 

49 

54097  9.73318 

122  9-73337 

J46  9-73357 
171  9-73377 
J95  9-73396 

84104  9.92482 
088  9.92473 
072  9.92465 
057  9.92457 
04I  9.92449 

64322  9.80836 
363  9.80864 
404  9.80892 
446  9.80919 
487  9.80947 

0.19164  1.5547 
0.19136   537 
0.19108   527 
0.19081   517 
O.I9053   507 

15 

14 
13 

12 
II 

50 

51 
52 
53 
54 

54220  9.73416 
244  9-73435 
269  9-73455 
293  9-73474 
3i7  9-73494 

84025  9.92441 

83994  9.92425 
978  9.92416 
962  9.92408 

64528  9.80975 
569  9.81003 
610  9.81030 
652  9.81058 
693  9.81086 

0.19025  1.5497 
0.18997   487 
0.18970   477 
0.18942   468 
0.18914   458 

10 

9 

8 

i 

55 

56 

f? 

r« 

54342  9.73513 
366  9-73533 
39i  9-73S52 
4i5  9-73572 
440  9-73591 
464  9.73611 

83946  9.92400 
930  9.92392 
915  9.92384 
899  9-92376 
883  9.92367 
867  9-92359 

64734  9.81113 
775  9.81141 
817  9.81169 
858  9.81196 
899  9.81224 
941  9.81252 

0.18887  1.5448 
0.18859  '438 
0.18831   428 
0.18804   418 
0.18776   408 
0.18748   399 

5 

4 
3 

2 

I 

0 

Nat.CoSLog.  d. 

Nat.  Sin  Log.  d. 

Nat.CotLog.|c.d. 

Log.Tan  Nat. 

f 

57° 


33C 


/ 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

T 

6 

7 
8 
9 

54464  9.73611 
488  9.73630 
5i3  9-73650 
537  9-73669 
561  9.73689 

19 

20 

19 
20 

19 
19 

2O 

19 
19 
20 

19 
19 
20 

19 
19 
20 
19 
19 
19 
19 
2O 
19 
19 
19 
19 
20 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 

18 
19 
19 
19 
19 
19 
18 

19 
19 
19 
18 

19 

83867  9.92359 
851  9-9235I 
835  9-92343 
819  9-92335 
804  9.92326 

8 
8 
8 
9 
8 
8 
8 

9 
8 

8 
8 

9 
8 
8 

9 
8 
8 
8 
9 
8 
8 

9 

8 
8 

9 
8 
8 

8 

9 
8 
8 
9 
8 

9 
8 
8 
9 
8 

9 
8 
8 
9 
8 

9 
8 

9 
8 

9 
8 

9 
8 

9 
8 

9 

8 

9 
8 

9 

64941  9.81252 

982  9.81279 
65024  9.81307 
065  9-81335 
106  9.81362 

3 

28 
27 
28 
28 

27 
28 

27 
28 
28 

27 
28 
27 
28 

27 

28 

27 
28 

27 

28 

27 
28 
27 
28 

27 
28 
27 
28 

27 

28 

27 
28 

27 

27 
28 

27 
28 

27 
27 
28 

27 
28 
27 

27 

28 
27 
27 
28 
27 

27 
28 

27 
27 
27 
28 

27 
27 
27 
28 

0.18748  1.5399 
0.18721  389 
0.18693  379 
0.18665  369 
0.18638  359 

60 

59 
58 
57 
56 

54586  9-73708 
610  9.73727 
635  9-73747 
659  9-73766 
683  9-73785 

83788  9.92318 
772  9.92310 
756  9.92302 
740  9.92293 
724  9.92285 

65148  9.81390 
189  9.81418 
231  9.81445 
272  9.81473  i 
314  9.81500 

0.18610  1.5350 
0.18582  340 

0-18555  330 
0.18527  320 
0.18500  311 

55 

54 
53 
52 
5i 
50" 
49 
48 

4£ 
46 

10 

ii 

12 
13 

14 

54708  9-73805 
732  9.73824 
756  9.73843 
781  9.73863 
805  9.73882 

83708  9.92277 
692  9.92269 
676  9.92260 
660  9.92252 
645  9.92244 

65355  9-81528 
397  9.8I556 
438  9.81583 
480  9.81611 
521  9.81638 

0.18472  1.5301 
0.18444  291 
0.18417  282 
0.18389  272 
0.18362  262 

15 

16 
J7 
18 
19 

54829  9.73901 
854  9-73921 
878  9-73940 
902  9-73959 
927  9-73978 

83629  9.92235 
613  9.92227 
597  9.92219 
581  9.92211 
565  9.92202 

65563  9.81666 
604  9.81693 
646  9.81721 
688  9.81748 
729  9.81776 

0.18334  1.5253 
0.18307  243 
0.18279  233 
0.18252  224 
0.18224  214 

45 

44 
43 
42 

4i 

20 

21 
22 

23 

24 

54951  9-73997 
975  9-740I7 
999  9-74036 
55024  9.74055 
048  9.74074 

83549  9.92194 
533  9-92I86 
517  9-92I77 
501  9.92169 
485  9.92161 

65771  9.81803 
813  9.81831 
854  9.81858 
896  9.81886 
938  9.81913 

0.18197  1.5204 
0.18169  195 
0.18142  185 
0.18114  175 
0.18087  166 

40 

39 
38 

1 

25 

26 
27 
28 
29 

55072  9.74093 
097  9.74113 
121  9.74132 

145  9-74I5I 
169  9.74170 

83469  9.92152 
453  9.92144 
437  9.92136 
421  9.92127 
405  9.92119 

65980  9.81941 
66021  9.81968 
063  9.81996 
105  9.82023 
147  9.82051 

0.18059  1.5156 
0.18032  147 
0.18004  137 
0.17977  127 
0.17949  118 

35 

34 
33 
32 
3i 

30 

3i 
32 
33 
34 

55194  9.74189 
218  9.74208 
242  9.74227 
266  9.74246 
291  9.74265 

83389  9-92111 
373  9-92102 
356  9.92094 
340  9.92086 
324  9.92077 

66189  9.82078 
230  9.82106 
272  9.82133 
314  9.82161 
356  9.82188 

0.17922  1.5108 
0.17894  099 
0.17867  089 
0.17839  080 
0.17812  070 

30 

29 
28 
27 
26 

35 

36 

P 

39 

553  J5  9-74284 
339  9-74303 
363  9-74322 
388  9.74341 
412  9.74360 

83308  9.92069 
292  9.92060 
276  9.92052 
260  9.92044 
244  9.92035 

66398  9.82215 
440  9.82243 
482  9.82270 
524  9.82298 
566  9.82325 

0.17785  1.5061 
0.17757  051 
0.17730  042 
0.17702  032 
0.17675  023 

25 

24 

23 

22 
21 

20 

19 

18 

% 

40 

4i 
42 

43 
44 

55436  9-74379 
460  9.74398 
484  9.74417 
509  9.74436 
533  9-74455 

83228  9.92027 

212  9.92018 
195  9.92010 
179  9.92002 
I63  9.91993 

66608  9.82352 
650  9.82380 
692  9.82407 
734  9.82435 
776  9.82462 

0.17648  1.5013 
0.17620  004 

0.17593  1-4994 
0.17565  985 

0.17538  975 

45 

46- 

47 
48 

49 

55557  9-74474 
581  9.74493 
605  9-74512 
630  9-74531 
654  9-74549 

83147  9.91985 
131  9.91976 
115  9.91968 
098  9.91959 
082  9.9I95I 

66818  9.82489 
860  9.82517 
902  9.82544 
944  9.82571 
986  9.82599 

0.17511  1.4966 
0.17483  957 
0.17456  947 
0.17429  938 
0.17401  928 

15 

14 
13 

12 
II 

50 

5* 
52 
53 
54 

55678  9.74568 
702  9.74587 
726  9.74606 
750  9.74625 
775  9.74644 

83066  9.91942 
050  9.91934 
034  9.91925 
017  9.9I9I7 

ooi  9.91908 

67028  9.82626 
071  9.82653 
113  9.82681 
155  9.82708 
197  9.82735 

0.17374  1.4919 
0.17347  9io 
0.17319  900 
0.17292  891 
0.17265  882 

10 

9 
8 

I 

55 

56 

1 
1 

55799  9.74662 
823  9.74681 
847  9-74700 
871  9.747J9 
895  9-74737 
919  9.74756 

82985  9.91900 
969  9.91891 

953  9.91883 
936  9.91874 
920  9.91866 
904  9.91857 

67239  9.82762 
282  9.82790 
324  9.82817 
366  9.82844 
409  9.82871 
45!  9.82899 

0.17238  14872 
0.17210  863 
0.17183  854 
0.17^56  844 
0.17129  835 
0.17101  826 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d.|Log.TanNat. 

/ 

56C 


34 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

55919  9-74756 
943  9-74775 
968  9.74794 
992  9.74812 
56016  9.74831 

ON  ONOO  ON  ON  OO  ON  ONOO  ON  OO  ON  ONOO  ON  OO  ONOO  ON  00  ONOO  ONOO  ON  OO  ONOO  OO  ON  OO  ONOO  CO  ONOOOOOO  ONOOOO  ONOO  OO  00  OO  ONOO  OO  OO  OO  OO  ONOO  OO  OO  OO  OO  OO  OO 

82904  9.91857 
887  9.91849 
871  9.91840 
855  9-91832 
839  9.91823 

8 

9 
8 

9 
8 

9 
8 

9 
8 

9 

9 
8 

9 
8 

9 

9 
9 

8 

9 
9 
8 

9 
9 
8 

9 
9 
8 
9 

9 
9 
9 
8 

9 
9 
8 
9 
9 
9 
8 

9 
9 

9 
9 
8 

9 
9 
9 
9 
8 

9 
9 
9 
9 
9 

67451  9.82899 
493  9.82926 
536  9.82953 
578  9.82980 
620  9.83008 

27 
27 
27 
28 
27 
27 

27 
27 
27 
27 
27 
28 

27 
27 
27 
27 

27 
27 
28 

27 
27 
27 
27 

27 
27 

27 
27 
27 

27 
28 

27 
27 
27 
27 
27 
27 
27 
27 

27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
26 

27 

27 

27 

27 
27 
27 
27 
27 
27 

0.17101  1.4826 
0.17074   816 
0.17047   807 
0.17020   798 
0.16992   788 

60 

it 
3 

5 

6 

7 
8 

9 
10 

ii 

12 
13 
14 

56040  9.74850 
064  9.74868 
088  9.74887 

112  9.74906 
136  9.74924 

82822  9.91815 
806  9.91806 
790  9.91798 
773  9-9!789 
757  9-9I78i 

67663  9.83035 
705  9-83062 
748  9.83089 
790  9.83117 
832  9.83144 

0.16965  1.4779 
0.16938   770 
0.16911   761 
0.16883   751 
0.16856   742 

55 

54 
53 
52 
5i 
50 
49 
48 

47 
46 

56160  9-74943 
I84  9.74961 
208  9.74980 

232  9-74999 
256  9-75017 

82741  9.91772 
724  9.91763 
708  9.91755 
692  9.91746 
675  9-9I738 

67875  9.83I7I 
917  9.83198 
960  9.83225 
68002  9.83252 
045  9.83280 

0.16829  1.4733 
0.16802   724 
0.16775   715 
0.16748   705 
0.16720   696 

15 

16 

17 
18 

19 
20 

21 
22 
23 

24 

56280  9.75036 
305  9-75054 
329  9-75073 
353  9-7509I 
377  9-75HO 

82659  9.91729 
643  9.91720 
626  9.91712 
610  9.91703 
593  9-91695 

68088  9.83307 

J3°  9-83334 
173  9.83361 
215  9-83388 
258  9-83415 

0.16693  1.4687 
0.16666   678 
0.16639   669 
0.16612   659 
0.16585   650 

45 

44 
43 
42 
4i 
40 

P 

% 

35 

34 
33 
32 
3i 

56401  9.75128 
425  9-75I47 
449  9-75I<>5 
473  9-75I84 
497  9-75202 

82577  9.91686 
561  9.91677 
544  9.91669 
528  9.91660 
511  9.91651 

68301  9.83442 
343  9-83470 
386  9.83497 
429  9-83524 
47i  9.83551 

0.16558  1.4641 
0.16530   632 
0.16503   623 
0.16476   614 
0.16449   605 

25 

26 

27 
28 

29 
80 

3i 
32 
33 
34 

56521  9.75221 
545  9-75239 
569  9-75258 
593  9-75276 
617  9.75294 

82495  9.91643 
478  9.91634 
462  9.91625 
446  9.91617 
429  9.91608 

68514  9.83578 
557  9-83605 
600  9.83632 
642  9.83659 
685  9.83686 

0.16422  1.4596 
0-16395   586 
0.16368   577 
0.16341   568 
0.16314   559 

56641  9.75313 
665  9-75331 
689  9.75350 
7X3  9-753^8 
736  9-75386 

82413  9.91599 
396  9.91591 
380  9.91582 
363  9.91573 
347  9-9I565 

68728  9.83713 
771  9.83740 
814  9.83768 

857  9-83795 
900  9.83822 

0.16287  1.4550 
0.16260   541 
0.16232   532 
0.16205   523 
0.16178   514 

30 

i 

35 

36 

P 

39 

56760  9.75405 
784  9.75423 
808  9.75441 
832  9-75459 
856  9-75478 

82330  9.91556 
314  9.91547 
297  9-9I538 
281  9.91530 
264  9.91521 

68942  9.83849 
985  9.83876 
69028  9.83903 
071  9.83930 
114  9.83957 

0.16151  1.4505 
0.16124   496 
0.16097   487 
0.16070   478 
0.16043   469 

25 

24 
23 

22 
21 

40 

41 
42 

43 

44 

56880  9.75496 
904  9-755I4 
928  9-75533 
952  9-75551 
976  9.75569 

82248  9.91512 
231  9.91504 
214  991495 
198  9.91486 
181  9.91477 

69157  9-83984 

200  9.84011 
243  9.84038 
286  9.84065 
329  9.84092 

0.16016  1.4460 
0.15989   451 
0.15962   442 

0.15935   433 
0.15908   424 

20 

19 
18 

3 

45 

46 

47 
48 

49 

57000  9.75587 
024  9.75605 
047  9-75624 
071  9.75642 
095  9-7566o 

82165  9.91469 
148  9.91460 
132  9.91451 
115  9.91442 
098  9.91433 

69372  9.84119 
416  9.84146 

459  9-84I73 
502  9.84200 
545  9.84227 

0.15881  1.4415 
0.15854   406 
0.15827   397 
0.15800   388 
0.15773   379 

15 

14 

13 

12 
II 

50 

5i 

52 
53 
54 

57119  9.75678 
143  975696 
167  9.75714 
J9i  9-75733 
215  9-75751 

82082  9.91425 
065  9.91416 
048  9.91407 
032  9.91398 
015  9-91389 

69588  9.84254 
631  9.84280 

675  9-84307 
718  9.84334 
761  9-84361 

0.15746  1.4370 
0.15720   361 

0.15693   SS2 
0.15666   344 
0.15639   335 

10 

7 
6 

55 

56 

H 
ft 

57238  9.75769 
262  9.75787 
286  9.75805 
310  9.75823 
334  9.75841 
358  9-75859 

81999  9.91381 
982  9.91372 
965  9-91363 
949  9-9I354 
932  9.91345 
915  9.91336 

69804  9.84388 
847  9-84415 
891  9.84442 

934  9-84469 
977  9.84496 
70021  9.84523 

0.15612  1.4326 
0.15585   3i7 
0.15558   308 
0.15531   299 
0.15504   290 
0.15477   281 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d. 

Log.TanNat. 

f 

35 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d.  Log.  Cot  Nat. 

0 

I 

2 

3 
4 

57358  9-75859 
381  9.75877 
405  9-75895 
429  9-759!3 
453  975931 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

17 
18 

18 
18 
18 
18 

17 
18 
18 
18 

17 
18 
18 
18 

17 
18 
18 

17 

18 
18 

17 
18 
18 
17 
18 
18 

17 
18 

17 
18 

17 

18 

17 
18 

17 
18 

J7 
18 

17 
18 

17 
18 
17 
17 

18 

81915  9.91336 
899  9-9x328 
882  9.91319 
865  9.91310 
848  9.91301 

8 
9 
9 
9 
9 
9 
9 
8 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
8 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

10 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

10 

9 
9 
9 
9 
9 
9 

10 

9 
9 

9 
9 

70021  9.84523 
064  9.84550 
107  9.84576 
151  9.84603 
194  9.84630 

27 
26 
27 

27 
27 
27 

27 
27 
26 
27 
27 
27 

27 
27 

26 
27 
27 

27 
27 
26 

27 
27 
27 
26 

27 
27 

2£ 
26 

27 
27 

27 
26 
27 
27 
26 

27 
27 
26 
27 
27 
26 
27 
27 
26 

27 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 

0.15477  1.4281 
0.15450  273 
0.15424  264 
0.15397  255 
0.15370  246 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

5 

6 

I 

9 

To 

ii 

12 
13 
14 

57477  9-75949 
501  9.75967 

524  9-75985 
548  9.76003 
572  9.76021 

81832  9.91292 
815  9.91283 
798  9.91274 
782  9.91266 
765  9.91257 

70238  9.84657 
281  9.84684 
325  9.84711 
368  9.84738 
412  9.84764 

0.15343  1.4237 
0.15316  229 

0.15289     220 
O.I5262    211 
0.15236     202 

57596  9.76039 
619  9.76057 
643  9.76075 
667  9.76093 
691  9.76111 

81748  9.91248 
731  9.91239 
714  9.91230 
698  9.91221 
681  9.91212 

70455  9-8479I 
499  9.84818 
542  9.84845 
586  9.84872 
629  9.84899 

0.15209  1.4193 
O.I5I82     185 
O.I5I55     176 
O.I5I28     167 
O.I5IOI     158 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

15 

16 

17 
18 

19 
20 

21 
22 
23 

24 

57715  9-76129 
738  9.76146 
762  9.76164 
786  9.76182 
810  9.76200 

81664  9.91203 
647  9.91194 
631  9.91185 
614  9.91176 
597  9.91167 

70673  9.84925 
717  9-84952 
760  9.84979 
804  9.85006 
848  9-85033 

0.15075  I.4I50 
0.15048     141 
O.I502I     132 
0.14994     124 
0.14967     115 

57833  9-762I8 
857  9.76236 
88  1  9.76253 
904  9.76271 
928  9.76289 

81580  9.91158 
563  9.91149 
546  9.91141 
530  9.91132 
513  9.91123 

70891  9.85059 
935  9-85086 
979  9.85113 
71023  9.85140 
066  9.85166 

O.I494I  I.4I06 
O.I49I4    097 
0.14887    089 
0.14860    080 
0.14834    071 

40 

39 
38 
37 
36 

25 

26 

27 
28 
29 
30 
3i 
32 
33 
34 

57952  9-76307 
976  9.76324 
999  9-76342 
58023  9.76360 
047  9-76378 

81496  9.91114 
479  9.91105 
462  9.91096 
445  9.91087 
428  9.91078 

71110  9.85193 
J54  9-85220 
198  9.85247 
242  9.85273 
285  9-85300 

0.14807  I.406j 
0.14780    054 
0.14753    045 
0.14727    037 
0.14700    028 

35 

34 
33 
32 
31 

58070  9.76395 
094  9.76413 
118  9.76431 
141  9.76448 
165  9.76466 

81412  9.91069 
395  9.91060 
378  9.91051 
361  9.91042 
344  9.91033 

71329  9.85327 
373  9-85354 
417  9-85380 
461  9.85407 

0.14673  I.40I9 

0.14646   on 

O.I4620    002 
0.14593  1.3994 
0.14566    985 

30 

27 
26 

35 

36 

P 

39 
40 

4i 
42 

43 

44 

58189  9.76484 

212  9.76501 
236  9.76519 
260  9.76537 
283  9-76554 

81327  9.91023 
310  9.91014 
293  9-9  J  005 
276  9.90996 
259  9.90987 

71549  9.85460 
593  9-85487 
637  9-855I4 
68  1  9.85540 

725  9.85567 

0.14540  1.3976 
O.I45I3    968 

0.14486   959 
0.14460   951 
0.14433   942 

25 

24 
23 

22 
21 

20 

19 
18 

17 
16 

58307  9.76572 
330  9-76590 

354  9-76607 
378  9.76625 
401  9.76642 

81242  9.90978 
225  9.90969 
208  9.90960 
191  9-9095I 
174  9.90942 

71769  9.85594 
813  9.85620 

857  9.85647 
901  9.85674 
946  9.85700 

0.14406  1.3934 
0.14380   925 
0-14353   9i6 
0.14326   908 
0.14300   899 

45 

46 

47 
48 

49 

58425  9.76660 
449  9.76677 
472  9.76695 
496  9.76712 
519  9.76730 

81157  9.90933 
140  9.90924 
123  9.90915 
106  9.90906 
089  9.90896 

71990  9.85727 
72034  9.85754 
078  9.85780 

122  9.85807 
167  9.85834 

0.14273  1.3891 
0.14246   882 
0.14220   874 
0.14193   865 
0.14166   857 

15 

H 
13 

12 
II 

50 

5i 

52 
53 
54 

58543  9-76747 
567  9-76765 
590  9.76782 
614  9.76800 
637  9.76817 

81072  9.90887 
055  9.90878 
038  9.90869 

021  9.90860 
004  9.90851 

722II  9.85860 
255  9.85887 
299  9.85913 

344  9-85940 
388  9.85967 

0.14140  1.3848 
0.14113   840 
0.14087   831 
0.14060   823 
0.14033   814 

10 

9 

8 

7 
6 

55 

56 
57 

58 

% 

58661  9.76835 
684  9.76852 
708  9.76870 
731  9.76887 
755  9-76904 
779  9-76922 

80987  9.90842 
970  9.90832 

953  9.90823 
936  9.90814 
919  9.90805 
902  9.90796 

72432  9.85993 
477  9.86020 
521  9.86046 
565  9.86073 
610  9.86100 
654  9.86126 

0.14007  1.3806 
0.13980   798 

0-13954   789 
0.13927   781 
0.13900   772 
0.13874   764 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.CotLog.|c.d. 

Log.Tan  Nat. 

t 

54 


36C 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

58779  9.76922 
802  9.76939 
826  9.76957 
849  9.76974 
873  9.76991 

17 

18 

17 
17 
18 
17 
17 
18 

17 
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17 

18 

17 
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17 
18 
17 
17 
17 
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17 
17 
17 
17 
17 
17 
17 
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17 
17 
17 
17 
17 
17 
17 
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17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
16 

17 
17 
17 
17 
17 
17 
16 

17 
17 
17 
17 
16 

80902  9.90796 
885  9.90787 
867  9.90777 
850  9.90768 
833  9.90759 

9 

10 

9 
9 
9 

9 
10 

9 
9 
9 

10 

9 
9 
9 

10 

9 
9 
9 

10 

9 
9 

10 

9 
9 
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10 

9 
9 
10 

9 
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10 

9 
10 

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9 

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9 
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10 

9 

10 

9 

10 

9 
9 

10 

9 

10 

9 

10 
9 

10 

9 
10 

9 

10 

9 

10 

9 

72654  9.86126 

699  9.86153 

743  9.86179 
788  9.86206 
832  9.86232 

27 
26 
27 
26 

27 
26 
27 
26 
27 
27 
26 

27 
26 
27 
26 

2£ 
26 

26 

27 
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26 

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26 
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26 
27 
26 
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2£ 
26 

26 
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0.13874  1.3764 
0.13847  755 
0.13821  747 
0.13794  739 
0.13768  730 

60 

59 
58 

11 

5 

6 

7 
8 

9 

58896  9.77009 

920  9.77026 

943  9-77043 
967  9.77061 
990  9.77078 

80816  9.90750 

799  9-9074I 
782  9.90731 
765  9.90722 
748  9-90713 

72877  9.86259 
921  9.86285 
966  9.86312 
73010  9.86338 
055  9-86365 

0.13741  1.3722 
0.13715  713 
0.13688  705 
0.13662  697 
0.13635  688 

55 

54 
53 
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SJ 

10 

ii 

12 
13 

14 

'16 

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17 
18 

19 

59014  9.77095 
°37  9-77H2 
061  9.77130 
084  9.77147 
108  9.77164 

80730  9.90704 
713  9.90694 
696  9.90685 
679  9.90676 
662  9.90667 

73100  9.86392 
144  9.86418 
189  9.86445 
234  9.86471 
278  9.86498 

0.13608  1.3680 
0.13582  672 
0.13555  663 
0.13529  655 
0.13502  647 

50 

49 
48 
47 
46 

59131  9.77181 

154  9.77199 
178  9.77216 

201  9.77233 
225  9.77250 

80644  9.90657 
627  9.90648 
6  10  9.90639 
593  9-90630 
576  9.90620 

73323  9-86524 
368  9.86551 
413  9.86577 
457  9.86603 
502  9.86630 

0.13476  1.3638 
0.13449  630 
0.13423  622 
0.13397  613 
0.13370  605 

45 

44 
43 
42 
4i 

20 

21 
22 

23 

24 

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26 

27 
28 
29 

59248  9.77268 
272  9.77285 
295  9-77302 
3l8  9.77319 
342  9.77336 

80558  9.90611 
541  9.90602 
524  9-90592 
507  9.90583 
489  9.90574 

73547  9-86656 
592  9.86683 
637  9.86709 

0.13344  1.3597 
0.13317  588 
0.13291  580 
0.13264  572 
0.13238  564 

40 

3 

1 

59365  9-77353 
389  9-77370 
412  9.77387 
436  9.77405 
459  9-77422 

80472  9.9056$ 
455  9-90555 
438  9-90546 
420  9-90537 
403  9-90527 

73771  9.86789 
816  9.86815 
861  9.86842 
906  9.86868 
951  9.86894 

0.13211  1.3555 
0.13185  547 

0-13158  539 
0.13132  531 
0.13106  522 

35 

34 
33 
32 
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30 

31 
32 
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59482  9-77439 
506  9.77456 
529  9-77473 
552  9.77490 
576  9-77507 

80386  9.90518 
368  9.90509 
351  9.90499 
334  9-90490 
316  9.90480 

73996  9.86921 
74041  9.86947 
086  9.86974 
131  9.87000 
176  9.87027 

0.13079  1.3514 
0.13053  5o6 
0.13026  498 
0.13000  490 
0.12973  481 

30 

3 
5 

35 

36 
37 
38 
39 
40 
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42 
43 
44 

59599  9.77524 
622  9.77541 
646  9-77558 
669  9-77575 
693  9-77592 

80299  9.90471 
282  9.90462 
264  9.90452 
247  9.90443 
230  9.90434 

74221  9.87053 
267  9.87079 
312  9.87106 
357  9-87I32 
402  9.87158 

0.12947  1.3473 
0.12921  465 
0.12894  457 
0.12868  449 
0.12842  440 

25 

24 
23 

22 
21 

59716  9.77609 
739  9-77626 
763  9.77643 
786  9.77660 
809  9.77677 

80212  9.90424 
195  9-9041$ 
178  9.90405 
160  9.90396 
143  9-90386 

74447  9-87185 
492  9.87211 
538  9.87238 
583  9.87264 
628  9.87290 

0.12815  1.3432 
0.12789  424 
0.12762  416 
0.12736  408 
0.12710  400 

20 

19 
18 

17 
16 

45 

46 

47 
48 

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59832  9.77694 

856  9-777I1 
879  9.77728 
902  9.77744 
926  9.77761 

80125  9.90377 
108  9.90368 
°9i  9-90358 
073  9-90349 
056  9-90339 

74674  9.87317 
719  9.87343 
764  9.87369 
810  9.87396 
855  9.87422 

0.12683  1.3392 
0.12657  384 
0.12631  375 
0.12604  367 
0.12578  359 

15 

14 
13 

12 
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50 

51 

52 
53 

54 

59949  9.77778 
972  9-77795 
995  9-778I2 
60019  9.77829 
042  9.77846 

80038  9.90330 

021  9.90320 
003  9.903II 
79986  9.90301 
968  9.90292 

74900  9.87448 
946  9.87475 
991  9.87501 
75037  9.87527 
082  9-87554 

0.12552  1.3351 
0.12525  343 
0.12499  335 
0.12473  327 
0.12446  319 

10 

7 
6 

55 

56 

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60065  9.77862 
089  9.77879 

112  9.77896 
135  9.779I3 
158  9-77930 
182  9-77946 

7995  J  9-90282 
934  9-90273 
916  9.90263 
899  9.90254 
88  1  9.90244 
864  9.90235 

75128  9.87580 
173  9.87606 
219  9.87633 

355  9-87711 

0.12420  1.3311 
0.12394  303 
0.12367  295 
0.12341  287 
0.12315  278 
0.12289  270 

5 

4 
3 

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0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.CotLog.|c.d.|Log.TanNat. 

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Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog.  c.d.  Log.  Cot  Nat. 

0 

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2 

3 

4 

60182  9.77946 
205  9-77963 
228  9.77980 

251  9-77997 
274  9.78013 

17 
17 

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17 

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17 
17 
16 

17 
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17 
17 
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17 
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17 
17 
16 
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17 
16 
17 
16 
16 

17 
16 

17 
16 
16 

17 
16 

17 
16 
16 

17 
16 
16 
16 
17 
16 
16 
17 
16 
16 
16 
16 

17 
16 
16 
16 

79864  9.90235 
846  9.90225 
829  9.90216 
811  9.90206 
793  9.90197 

10 

9 
10 

9 

10 

9 

10 

9 

10 
10 

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10 

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10 
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10 
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10 
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10 
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10 
10 
10 
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401  9.87738 

447  9-87764 
492  9.87790 
538  9.87817 

27 
26 
26 
27 
26 
26 
26 

27 
26 
26 

26 

27 
26 
26 
26 

26 
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26 
26 
26 
26 
27 
26 
26 
26 
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27 
26 
26 
26 
26 
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27 
26 
26 
26 
26 
26 
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26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 

0.12289  1.3270 
0.12262   262 
0.12236   254 

O.I22IO    246 
O.I2I83    238 

60 

59 
58 
57 
56 

5 

6 

7 
8 

9 

60298  9.78030 
321  9.78047 
344  9.78063 
367  9.78080 
390  9.78097 

79776  9.90187 
758  9.90178 
741  9.90168 
723  9.90159 
706  9.90149 

75584  9.87843 
629  9.87869 
675  9.87895 
721  9.87922 
767  9.87948 

O.I2I57  1.3230 
O.I2I3I    222 
O.I2IOg    214 
0.12078    206 
O.I2052    198 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 
14 

60414  9.78113 
437  9-78I30 
460  9.78147 
483  9.78163 
506  9.78180 

79688  9.90139 
671  9.90130 
653  9.90120 
635  9.90111 
618  9.90101 

75812  9.87974 
858  9.88000 
904  9.88027 
950  9.88053 
996  9.88079 

O.I2O26  1.3190 
0.12000    182 
O.II973    175 
O.II947    167 
O.II92I    159 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

60529  9.78197 
553  9-78213 
576  9.78230 
599  9.78246 
622  9.78263 

79600  9.90091 
583  9.90082 
565  9.90072 
547  9.90063 
530  9.90053 

76042  9.88105 
088  9.88131 
134  9.88158 
180  9.88184 
226  9.88210 

0.11895  1.3151 
O.II869    143 
O.II842    135 

0.11816   127 
0.11790   119 

45 

44 
43 

42 

4i 

20 

21 
22 

23 
24 

60645  9.78280 
668  9.78296 
691  9.78313 
714  9-78329 
738  9.78346 

79512  9.90043 
494  9-90034 
477  9.90024 
459  9,90014 
441  9.90005 

76272  9.88236 
318  9.88262 
364  9.88289 
410  9.88315 
456  9.8834! 

0.11764  1.3111 
0.11738   103 
0.11711   095 
0.11685   087 
0.11659   079 

40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

25 

26 
27 
28 
29 

60761  9.78362 
784  9.78379 
807  9.78395 
830  9.78412 
853  9.78428 

79424.9.89995 
406  9.89985 
388  9*89976 
371  9.89966 
353  9.89956 

76502  9.88367 
548  9-88393 
594  9.88420 
640  9.88446 
686  9.88472 

0.11633  1.3072 
0.11607   064 
0.11580   056 
0.11554   048 
0.11528   040 

30 

3i 
32 
33 
34 
35^ 
36 

P 

39 

60876  9.78445 
899  9.78461 
922  9.78478 
945  9.78494 
968  9.78510 

79335  9.89947 
318  9.89937 
300  9.89927 
282  9.89918 
264  9.89908 

76733  9.88498 
779  9.88524 
825  9.88550 
871  9.88577 
•  918  9.88603 

0.11502  1.3032 
0.11476   024 
0.11450   017 
0.11423   009 
0.11397   ooi 

30 

29 
28 
27 
26 

60991  9.78527 
61015  9.78543 
038  9.78560 
061  9.78576 
084  9.78592 

79247  9.89898 
229  9.89888 

211  9.89879 
I93  9.89869 
I76  9.89859 

76964  9.88629 
77010  9.88655 
057  9.88681 
103  9.88707 
149  9-88733 

0.11371  1.2993 
0.11345  985 

0.11319   977 
0.11293   970 
0.11267   962 

25 

24 

23 

22 
21 

40 

41 
42 
43 
44 
45" 
46 

47 
48 

49 

61107  9.76009 
130  9.78625 
153  9-78642 
176  9.78658 
199  9.78674 

79158  9.89849 

140  9.89840 

122  9.89830 
105  9.89820 
087  9.89810 

77196  9.88759 
242  9.88786 
289  9.88812 

0.11241  1.2954 
0.11214   946 
0.11188   938 
0.11162   931 
0.11136   923 

20 

19 
18 

17 
16 

61222  9.76091 

245  9-78707 
268  9.78723 
291  9.78739 
314  9.787^6 

79069  9.89801 
051  9.89791 
033  9.89781 

016  9.89771 
78998  9.89761 

77428  9.88890 
475  9.88916 
521  9.88942 
568  9.88968 
615  9.88994 

o.iiiio  1.2915 
0.11084   907 
0.11058   900 
0.11032   892 
0.11006   884 

15 

14 
J3 

12 
II 

50 

5i 

52 
53 
54_ 
55 

56 
57 
58 

| 

6i337  9-78772 
360  9.78788 
383  9.78805 
406  9.78821 
429  9.78837 

78980  9.89752 
962  9.89742 

944  9.89732 
926  9.89722 
908  9.89712 

77661  9.89020 
708  9.89046 
754  9.89073 
801  9.89099 
848  9-89125 

0.10980  1.2876 

0.10954  869 

0.10927   861 

0.10901   853 
0.10875   846 

10 

1 
i 

61451  9.78853 
474  9.78869 
497  9.78886 
520  9.78902 
543  9.78918 
566  9.78934 

78891  9.89702 
873  9.89693 
855  9.89683 

837  9.89673 
819  9.89663 
801  9-89653 

77895  9.89151 
941  9.89177 
988  9.89203 
78035  9.89229 
082  9.89255 
129  9.89281 

0.10849  1.2838 
0.10823   830 
0.10797   822 
0.10771   815 
0.10745   807 
0.10719   799 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d. 

Log.TanNat. 

f 

52° 


38C 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.  Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

61566  9.78934 
589  9.78950 
612  9.78967 
635  9-78983 
658  9.78999 

16 

17 
16 
16 
16 
16 
16 
16 
16 
16 
16 

17 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
15 
16 
16 
16 
16 
16 
16 
16 
16 
16 
15 
16 
16 
16 
16 
16 

IS 
16 
16 
16 
15 
16 
16 
16 

15 
16 

16 

15 
16 
16 
15 
16 
16 

15 
16 
16 

15 

78801  9.89653 
783  9-89643 
765  9.89633 
747  9.89624 
729  9.89614 

10 

IO 

9 

10 
10 
IO 
IO 
IO 
IO 
IO 
10 
10 
10 
10 
10 

9 

10 
10 
10 
10 
IO 
IO 
IO 
IO 
10 
10 
10 
10 

II 

IO 
10 
10 
10 
10 
IO 
10 
10 
10 
10 
10 
IO 

II 

IO 
10 
10 
IO 
IO 
IO 

II 

10 
IO 
IO 
IO 
10 

II 

IO 
IO 
10 

II 

10 

78129  9.89281 
175  9-89307 

222  9.89333 
269  9.89359 
3l6  9.89385 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 
26 
26 
26 

25 
26 
26 
26 

0.10719  1.2799 
0.10693   792 
0.10667   784 
0.10641   776 
0.10615   769 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 
48 

47 
46 

45 

44 
43 

42 
4i 

5 

6 

7 
8 

9 

61681  9.79015 
704  9.79031 
726  9.79047 
749  9.79063 
772  9-79079 

78711  9.89604 
694  9.89594 
676  9.89584 
658  9-89574 
640  9-89564 

78363  9.894II 
410  9.89437 

457  9-89463 
504  9.89489 
551  9.89515 

0.10589  1.2761 
0.10563   753 
0-10537   746 
0.10511   738 
0.10485   731 

10 

ii 

12 
13 
14 
IT 

16 

17 
18 

19 

61795  9-79095 
818  9.79111 
841  9.79128 
864  9.79144 
887  9.79160 

78622  9.89554 
604  9.89544 
586  9.89534 
568  9.89524 
550  9-895I4 

78598  9.89541 

645  9-89567 
692  9.89593 
739  9.89619 
786  9.89645 

0.10459  1.2723 
0.10433   715 
0.10407   708 
0.10381   700 
0-10355   693 

61909  9.79176 
932  9.79192 
955  9.79208 
978  9.79224 
62001  9.79240 

78532  9.89504 
514  9.89495 
496  9.89485 
478  9.89475 
460  9.89465 

78834  9.89671 
88  1  9.89697 
928  9.89723 

975  9.89749 
79022  9.89775 

0.10329  1.2685 
0.10303   677 
0.10277   670 
0.10251   662 
0.10225   655 

20 

21 

22 
23 
24 

2T 

26 

27 

28 

29 

62024  9-79256 
046  9.79272 
069  9.79288 
092  9.79304 

us  9-79319 

78442  9.89455 
424  9.89445 
405  9.89435 
387  9.89425 
369  9.89415 

79070  9.89801 
117  9.89827 
•  164  9.89853 
212^9.89879 
259*9.89905 

0.10199  1.2647 
0.10173   640 
0.10147   632 

O.IOI2I    624 

0.10095   617 

40 
P 

1 
35 

34 
33 
32 
3i 

62138  9.79335 
160  9-79351 
183  9-79367 
206  9.79383 
229  9-79399 

78351  9.89405 

333  9-89395 
315  9.89385 

297  9.89375 
279  9-89364 

79306  9.89931 

354  9.89957 
401  9.89983 
449  9.90009 
496  9.90035 

0.10069  1.2609 
0.10043   602 
0.10017   594 
0.09991   587 
0-09905   579 

30 

3i 
32 
33 

34 

62251  9.79415 
274  9.79431 
297  9-79447 
320  9.79463 
342  9.79478 

78261  9.89354 
243  9.89344 

225  9.89334 
206  9.89324 
188  9.89314 

79544  9.90061 
591  9.90086 
639  9.90112 
686  9.90138 
734  9.90164 

0.09939  1.2572 
0.09914   564 
0.09888   557 
0.09862   549 
0.09836   542 

30 

27 
26 

35 

36 
37 
38 
39 

62365  9.79494 
388  9.79510 
411  9.79526 
433  9-79542 
456  9-79558 

78170  9.89304 
152  9.89294 
134  9.89284 
116  9.89274 
098  9.89264 

79781  9.90190 
829  9.90216 
877  9.90242 
924  9.90268 
972  9.90294 

0.09810  1.2534 
0.09784   527 
0.09758   519 
0.09732   512 
0.09706   504 

25 

24 
23 

22 
21 

40 

41 
42 
43 

44 

62479  9.79573 
502  9.79589 
524  9.79605 
547  9-7962I 
570  9.79636 

78079  9.89254 
061  9.89244 
043  9.89233 
025  9.89223 
007  9.89213 

80020  9.90320 
067  9.90346 

US  9.9037I 
163  9.90397 

211  9.90423 

0.09680  1.2497 
0.09654   489 
0.09629   482 
0.09603   475 
0.09577   467 

20 

19 
18 

17 
16 

45 
46 

47 
48 

49 

62592  9.79652 
615  9.79668 
638  9.79684 
660  9.79699 
683  9.79715 

77988  9.89203 
970  9.89193 
952  9.89183 

934  9.89I73 
916  9.89162 

80258  9.90449 
306  9.90475 

354  9.90501 
402  9.90527 
450  9-90553 

0.09551  1.2460 
0.09525   452 
0.09499   445 
0-09473   437 
0.09447   430 

15 

14 

13 

12 

II 

10" 

9 

8 

I 

50 

5i 

52 

53 
54 

62706  9.79731 
728  9.79746 
75i  9-79762 
774  9-79778 
796  9.79793 

77897  9-89I52 
879  9.89142 
861  9.89132 
843  9.89122 
824  9.89112 

80498  9.90578 
546  9.90604 
594  9.90630 
642  9.90656 
690  9.90682 

0.09422  1.2423 
0.09396   415 
0.09370   408 
0.09344   401 
0.09318   393 

55 

56 

P 
6590 

62819  9.79809 
842  9.79825 
864  9.79840 
887  9.79856 
909  9.79872 
932  979887 

77806  9.89101 
788  9.89091 
76q  9.89081 

751  9.89071 

733  9-89060 
715  9.89050 

80738  9.90708 
786  9.90734 

834  9-90759 
882  9.90785 
930  9.90811 
978  9.90837 

0.09292  1.2386 
0.09266   378 
0.09241   371 
0.09215   364 
0.09189   356 
0.09163   349 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.Tan  Nat.|  t 

51 


39C 


t 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

:.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

62932  9.79887 

955  9-799°3 
977  9.79918 
63000  9.79934 

022  9.79950 

16 

15 
16 
16 

15 

16 

11 

15 
16 
15 
16 
15 
16 

15 
16 
15 
IS 
16 

15 
16 

15 
16 

15 
15 
16 

IS 
15 
16 

15 
15 
16 
15 
15 
16 

15 
IS 
15 
16 

15 
15 

\l 

15 
15 
15 
15 
15 
16 
15 
15 
15 
15 
15 
15 
15 
16 
15 
15 
15 

77715  9.89050 
696  9.89040 
678  9.89030 
660  9.89020 
641  9.89009 

10 
10 
10 

II 

10 
IO 

II 

IO 
10 
IO 

II 

10 
10 

II 

IO 
10 

II 

IO 
10 

II 

10 
10 

II 

10 
10 

II 

10 

II 

10 
10 

II 

10 

II 

10 

II 

IO 
IO 

II 

10 

II 

10 

II 

10 

II 

10 

II 

10 

II 

IO 

II 

10 

II 
II 

10 

II 

10 

II 

10 

II 
II 

80978  9.90837 
81027  9-90863 
075  9.90889 
123  9.90914 
171  9.90940 

26 
26 

25 
26 

26 
26 
26 

3 

26 
26 
26 

2S 
26 

26 
26 
26 

2 

26 
26 

25 
26 
26 
26 

25 
26 

26 
26 

25 
26 
26 
26 
25 
26 
26 
26 

25 
26 
26 

25 
26 
26 
26 

25 
26 
26 

25 
26 
26 

25 
26 
26 

25 

26 
26 

25 

26 
26 
25 

0.09163  1.2349 
0.09137  342 
0.09111  334 
0.09086  327 
0.09060  320 

60 

1 

5 

6 

I 

9 

63045  9.79965 
068  9.79981 
090  9.79996 

113  9.80012 
135  9.80027 

77623  9.88999 
605  9.88989 

550  9'.88958 

81220  9.90966 
268  9.90992 
316  9.91018 
364  9.91043 
413  9.91069 

0.09034  1.2312 
0.09008  305 
0.08982  298 
0.08957  290 
0.08931  283 

55 

54 
53 
52 
Si 

10 

II 

12 

13 
14 

63158  9.80043 

180  9.80058 
203  9.80074 
225  9.80089 
248  9.80105 

77531  9.88948 
513  9.88937 
494  9.88927 
476  9.88917 
458  9.88906 

81461  9.91095 
510  9.91121 
558  9.91147 
606  9.91172 
655  9.91198 

0.08905  1.2276 
0.08879  268 
0.08853  261 
0.08828  254 
0.08802  247 

50 

49 
48 
47 
46 

15 

16 

17 
18 

19 
20 

21 
22 

23 
24 

63271  9.80120 
293  9.80136 
316  9.80151 
338  9.80166 
361  9.80182 

77439  9.88896 
421  9.88886 
402  9.88875 

81703  9.91224 
752  9.91250 
800  9.91276 
849  9.91301 
898  9.91327 

0.08776  1.2239 
0.08750  232 
0.08724  225 
0.08699  218 

0.08673  210 

45 

44 
43 
42 

4i 

63383  9-80I97 
406  9.80213 
428  9.80228 
451  9.80244 
473  9-80259 

77347  9-88844 
329  9.88834 
310  9.88824 
292  9.88813 
273  9.88803 

81946  9.91353 
995  9.91379 
82044  9-9J404 
092  9.91430 
141  9.91456 

0.08647  1.2203 
0.08621  196 
0.08596  189 

0.08570  181 
0.08544  174 

40 

39 
38 
37 
36 

25 

26 

27 
28 
29 

63496  9.80274 
518  9.80290 
540  9.80305 
563  9.80320 
585  9.80336 

77255  9.88793 
236  9.88782 
218  9.88772 
199  9.88761 
181  9.88751 

82190  9.91482 
238  9.91507 
287  9-9I533 
336  9-9I559 
385  9.91585 

0.08518  1.2167 
0.08493  160 
0.08467  153 
0.08441  145 
0.08415  138 

35 

34 
33 
32 
3i 

30 

31 
S2 
33 
34 
35 
36 

P 

39 
40 

41 
42 
43 
44 

63608  9.80351 
630  9.80366 
653  9-80382 
675  9-80397 
698  9.80412 

77162  9.88741 
144  9.88730 
125  9.88720 

82434  9.91610 
483  9.91636 
531  9.91662 
580  9.91688 
629  9.91713 

0.08390  1.2131 
0.08364  124 
0.08338  117 
0.08312  109 

O.O8287  102 

30 

29 
28 
27 
26 

63720  9.80428 
742  9.80443 
765  9.80458 
787  9.80473 
8  10  9.80489 

77070  9.88688 
051  9.88678 
033  9.88668 
014  9-88657 
76996  9.88647 

82678  9.91739 
727  9.91765 
776  9.91791 
825  9.91816 
874  9.91842 

0.08261  1.2095 
0.08235  088 
0.08209  08  1 
0.08l84  074 
O.O8l58  066 

25 

24 
23 

22 
21 

63832  9.80504 
854  9.80519 

877  9-80534 
899  9.80550 
922  9.80565 

76977  9-88636 
959  9.88626 
940  9.88615 
921  9.88605 
903  9.88594 

82923  9.91868 
972  9.91893 
83022  9.91919 
071  9.91945 

120  9.91971 

0.08132  1.2059 
0.08107  052 
0.08081  045 
0.08055  038 
0.08O29  031 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

63944  9.80580 
966  9.80595 
989  9.80610 
64011  9.80625 
033  9.80641 

76884  9.88584 
866  9.88573 
847  9.88563 
828  9.88552 
8  10  9.88542 

83169  9.91996 

218  9.92022 
268  9.92048 

317  9.92073 
366.9.92099 

O.O8004  1.2024 
0.07978  017 
0.07952  009 
0.07927  002 
0.07901  I.I995 

15 

14 
13 

12 
II 

50 

5i 
52 
53 
54 
55 
56 
57 
58 

& 

64056  9.80656 
078  9.80671 
100  9.80686 
123  9.80701 
145  9.80716 

76791  9.88531 
772  9.88521 
754  9.88510 

717  9.88489 

83415  9.92125 
465  9.92150 
514  9.92176 

564  9.92202 

613  9.92227 

0.07875  I.I988 
0.07850  981 

0.07824  974 
0.07798  967 
0-07773  960 

10 

9 
8 

7 
6 

T 

4 

I 
0 

64167  9.80731 
190  9.80746 

212  9.80762 
234  9.80777 
256  9.80792 
279  9.80807 

76698  9.88478 
679  9.88468 
661  9-88457 
642  9.88447 
623  9.88436 
604  9-88425 

83662  9.92253 
712  9.92279 
761  9.92304 

8  ii  9.92330 
860  9.92356 
910  9.92381 

0.07747  1.1953 
0.07721  946 
0.07696  939 
0.07670  932 
0.07644  925 
0.07619  918 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d 

Nat.  Cot  Log 

c.d 

Log.TanNat 

t 

50C 


40° 


'  |Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.  Tan  Log 

c.d 

Log.  Cot  Nat 

0 

I 

2 

3 
4 
T 

6 

7 
8 

9 

64279  9.80807 
301  9.80822 

323  9.80837 
346  9.80852 
368  9.80867 

15 
15 
15 
15 
15 
15 
15 
15 
15 
IS 
15 
15 
IS 
15 
15 
15 
14 
15 
15 
15 
15 
15 
15 
15 
14 
15 
15 
15 
15 
14 
15 
15 
IS 
15 
14 
15 
15 
14 
15 
15 
IS 
14 
15 
15 
14 
15 
15 
14 
15 
15 
14 
15 
14 
15 
15 
14 
15 
14 
15 
14 

76604  9.88425 
586  9.88415 
567  9.88404 
548  9.88394 
530  9.88383 

10 
ii 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 
II 

10 

II 
II 
II 

10 

II 
II 
II 

10 

II 
II 
II 
II 

10 

II 
II 
II 
II 
II 
II 

10 

II 
II 
II 
II 
II 
II 
II 
II 

10 

II 
II 
II 
II 
II 
II 
II 
II 
II 

83910  9.92381 
960  9.92407 
84009  9.92433 
059  9.92458 
108  9.92484 

26 
26 

25 
26 

26 

Ii 

26 
25 
26 

2I 
26 

26 
25 
26 
26 

25 
26 

25 
26 
26 

25 
26 

25 
26 
26 

25 
26 

25 
26 

3 

26 
25 
26 

1 
3 

26 

3 

25 
26 

25 
26 

25 
26 
26 

25 
26 

25 
26 

25 
26 

25 
26 

2I 
26 

25 

0.07619  1.1918 

0-07593   91° 
0.07567   903 
0.07542   896 
0.07516   889 

60 

59 
58 
57 
56 
55 
54 
53 
52 
Si 
50 

s 

47 
46 

64390  9.80882 

412  9.80897 

435  9-80912 
457  9-80927 
479  9.80942 

76511  9.88372 
492  9.88362 
473  9-88351 
455  9-88340 
436  9.88330 

84158  9.92510 
208  9-92535 
258  9.92561 
307  9-92587 
357  9-92612 

0.07490  1.1882 
0.07465   875 
0.07439   868 
0.07413   861 
0.07388   854 

10 

ii 

12 
13 
14 

64501  9.80957 
524  9.80972 
546  9.80987 
568  9.81002 
590  9.81017 

76417  9.88319 
398  9.88308 
380  9.88298 
361  9.88287 
342  9.88276 

84407  9.92638 
457  9.92663 
507  9.92689 
556  9.92715 
606  9.92740 

0.07362  1.1847 
0-07337   840 
0.07311   833 
0.07285   826 
0.07260   819 

15 

16 

17 
18 

19 

64612  9.81032 
635  9.81047 
657  9.81061 
679  9.81076 
701  9.81091 

76323  9.88266 
304  9.88255 
286  9.88244 
267  9.88234 
248  9.88223 

84656  9.92766 
706  9.92792 
756  9.92817 
806  9.92843 
856  9.92868 

0.07234  1.1812 
0.07208   806 
0.07183   799 
0.07157   792 
0.07132   785 

45 

44 
43 
42 
41 

20 

21 
22 

23 
24 

25~ 

26 
27 
28 
29 

64723  9.81106 
746  9.81121 
768  9.81136 
790  9.81151 
812  9.81166 

76229  9.88212 

210  9.88201 
192  9.88191 
173  9.88180 
154  9.88169 

84906  9.92894 
956  9.92920 
85006  9.92945 

°57  9-9297i 
107  9.92996 

0.07106  1.1778 
0.07080   771 
0.07055   764 
0.07029   757 
0.07004   750 

40 

39 
38 

H 

64834  9.81180 
856  9.81195 
878  9.81210 
901  9.81225 
923  9.81240 

76135  9.88158 

116  9.88148 
097  9.88137 
078  9.88126 
059  9.88115 

85I57  9-93022 
207  9.93048 

257  9-93073 
308  9.93099 
358  9-93124 

0.06978  1.1743 
0.06952   736 
0.06927   729 
0.06901   722 
0.06876   715 

35 

34 
33 
32 
3i 
30 
29 
28 

2? 

25 

24 
23 

22 
21 

20 

19 
18 

11 
15 

14 
J3 

12 

II 

TO 

9 

8 

I 

30 

31 
32 
33 

34 

64945  9.81254 
967  9.81269 
989  9.81284 
65011  9.81299 
033  9.81314 

76041  9.88105 

022  9.88094 
003  9.88083 
75984  9.88072 
965  9.88061 

85408  9.93150 
458  9.93175 
509  9.93201 
559  9-93227 
609  9-93252 

0.06850  1.1708 
0.06825   702 
0.06799   695 
0.06773   688 
0.06748   681 

35 

36 
37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 

49 
50 

Si 

52 
53 
54 

65°55  9-81328 
077  9.81343 
loo  9.81358 

122  9.81372 
144  9.81387 

75946  9.88051 
927  9.88040 
908  9.88029 
889  9.88018 
870  9.88007 

85660  9.93278 
7io  9-93303 
761  9.93329 

0.06722  1.1674 
0.06697   667 
0.06671   660 
0.06646   653 
0.06620   647 

65166  9.81402 

188  9.81417 
210  9.81431 
232  9.81446 
254  9.81461 

75851  9.87996 
832  9.87985 
813  9-87975 

794  9.87964 
775  9.87953 

85912  9.93406 
963  9.93431 
86014  9-93457 
064  9.93482 
"5  9-93508 

0.06594  1.1640 
0.06569   633 
0.06543   626 
0.06518   619 
0.06492   612 

65276  9.81475 
298  9.81490 
320  9.81505 
342  9.81519 
364  9.81534 

75756  9.87942 
738  9-8793I 
719  9.87920 
700  9.87909 
680  9.87898 

86166  9.93533 
216  9.93559 
267  9-93584 
318  9.93610 
368  9.93636 

0.06467  1.  1  606 
0.06441   599 
0.06416   592 
0.06390   585 
0.06364   578 

65386  9.81549 
408  9.81563 
430  9.81578 
452  9-81592 
474  9.81607 

75661  9.87887 
642  9.87877 
623  9.87866 
604  9.87855 
585  9.87844 

86419  9.93661 
470  9.93687 
521  9-93712 
572  9-93738 
623  9-93763 

0.06339  1.1571 
0.06313   565 
0.06288   558 
0.06262   551 
0.06237   544 

55 

56 

P 

I9o 

65496  9.81622 
518  9.81636 
540  9.81651 
562  9.81665 
584  9.81680 
606  9.81694 

75566  9.87833 
547  9.87822 
528  9.87811 
509  9.87800 
490  9.87789 
471  9-87778 

86674  9.93789 
725  9-93814 
776  9.93840 
827  9-93865 
878  9.93891 
929  9-939J6 

0.06211  1.1538 
0.06186   531 
0.06160   524 
0-06135   517 
0.06109   510 
0.06084   504 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

/ 

49° 


41C 


'  |Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

65606  9.81694 
628  9.81709 
650  9.81723 
672  9.81738 
694  9.81752 

15 
14 
15 
14 
15 
14 
15 
14 
i5 
14 
15 
14 
14 
15 
14 
15 
14 
15 
14 
14 
i5 
14 
14 
15 
14 
14 
15 
14 
14 
14 
15 
14 
H 
15 
14 
14 
H 
14 
15 
14 
14 
14 
14 
15 
14 
14 
14 
14 
14 
14 
M 
15 
14 
14 
14 
14 
14 
14 
14 
14 

75471  9.87778 

433  9.87756 
414  9.87745 
395  9.87734 

ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 
ii 

12 
II 
II 
II 
II 
II 
II 
II 
II 
12 
II 
II 
II 
II 
II 
12 
II 
II 
II 
II 
12 
II 
II 
II 
II 
12 
II 
II 
12 
II 
II 
II 
12 
II 
II 
12 
II 
II 
12 
II 
II 
12 
II 
II 
12 

86929  9.93916 
980  9.93942 
87031  9-93967 
082  9.93993 
133  9.94018 

26 

25 
26 

25 
26 

% 
3 

25 
26 

2| 
26 

25 
26 

11 
11 

25 
26 

11 

25 
26 

25 

3 

25 
26 

25 
26 

2| 
26 

25 
26 
25 

3 

25 
26 

25 
26 

25 
25 
26 

25 
26 

25 
26 

25 
25 
26 

25 
26 

3 

25 
25 
26 

0.06084  I'I5°4 

0.06058  497 
0.06033  490 
0.06007  483 
0.05982  477 

60 

i 
I 

5 

6 

7 
8 

9 
TO 

ii 

12 
13 
14 

65716  9.81767 
738  9.81781 
759  9-8I796 
781  9.81810 
803  9.81825 

75375  9.87723 
356  9.87712 

337  9.87701 
318  9.87690 
299  9.87679 

87184  9.94044 
236  9.94069 
287  9.94095 
338  9.94120 
389  9.94146 

0.05956  1.1470 
0.05931  463 
0.05905  456 
0.05880  450 
0.05854  443 

55 

54 
53 
52 
5i 
50 

49 
48 

47 
46 

65825  9.81839 
847  9.81854 
869  9.81868 
891  9.81882 
913  9.81897 

75280  9.87668 
261  9.87657 
241  9.87646 

222  9.87635 
203  9.87624 

87441  9.94171 
492  9.94197 

543  9.94222 
595  9-94248 
646  9.94273 

0.05829  1.1436 
0.05803  430 
0.05778  423 
0.05752  416 
0.05727  410 

15 

16 

17 
18 
19 

65935  9-81911 
956  9.81926 
978  9.81940 
66000  9.81955 

022  9.81969 

75184  9.87613 
165  9.87601 
I46  9.87590 

87698  9.94299 
749  9.94324 
801  9.94350 
852  9.94375 
904  9.94401 

0.05701  1.1403 
0.05676  396 
0.05650  389 
0.05625  383 
0.05599  376 

45 

44 

43 
42 
41 

20 

21 
22 

23 
24 

25 

26 

29 
W 

31 
32 

33 
34 

66044  9.81983 
066  9.81998 
088  9.82OI2 
109  9.82026 
131  9.82041 

75088  9.87557 
069  9.87546 
050  9-87535 
030  9.87524 

on  9.87513 

87955  9.94426 
88007  9.94452 

059  9-94477 
no  9.94503 
162  9.94528 

0.05574  1.1369 
0.05548  363 
0.05523  356 

0.05497  349 
0.05472  343 

40 

39 

38 

1 
35 

34 
33 
32 
3i 
30 

29 
28 

3 

25 

24 
23 

22 
21 

66153  9.82055 
175  9.82069 
197  9.82084 

21  8  9.82098 

240  9.82II2 

74992  9.87501 

973  9-87490 
953  9.87479 
934  9.87468 
915  9.87457 

88214  9.94554 
265  9-94579 
317  9.94604 
369  9.94630 
421  9.94655 

0.05446  1.1336 
0.05421  329 
0.05396  323 
0.05370  316 
0.05345  310 

66262  9.82126 
284  9.82141 
306  9.82155 
327  9.82169 

349  9.82184 

74896  9.87446 
876  9.87434 

857  9.87423 
838  9.87412 
818  9.87401 

88473  9.9468i 
524  9-94706 
576  9.94732 
628  9.94757 
680  9.94783 

0.05319  1.1303 
0.05294  296 
0.05268  290 
0.05243  283 
0.05217  276 

35 

36 

P 

39 
40 

4i 
42 
43 
44 

66371  9.82198 
393  9.82212 
414  9.82226 
436  9.82240 
458  9.82255 

74799  9-87390 
780  9.87378 
760  9.87367 
741  9.87356 
722  9.87345 

88732  9.94808 
784  9.94834 
836  9.94859 
888  9.94884 
940  9.94910 

0.05192  1.1270 
0.05166  263 
0.05141  257 
0.05116  250 
0.05090  243 

66480  9.82269 
501  9.82283 
523  9.82297 
545  9.82311 
566  9.82326 

74703  9.87334 
683  9.87322 
664  9.87311 
644  9.87300 
625  9.87288 

88992  9.94935 
89045  9.94961 
097  9.94986 
149  9.95012 

201  9.95037 

0.05065  1.1237 
0.05039  230 
0.05014  224 
0.04988  217 

0.04963  211 

20 

19 
18 

| 

45 

46 

47 
48 

49 
50 

Si 
52 
53 
54 

66588  9.82340 
610  9.82354 
632  9.82368 
653  9-82382 
675  9.82396 

74606  9.87277 
586  9.87266 
567  9.87255 
548  9.87243 
528  9.87232 

89253  9-95062 
306  9.95088 
358  9.95II3 
4IO  9-95I39 
463  9.95164 

0.04938  I.I204 
0.04912  197 
0.04887  191 
0.04861  184 
0.04836  178 

15 

14 
13 

12 
II 

66697  9.82410 
718  9.82424 
740  9.82439 
762  9.82453 
783  9.82467 

74509  9.87221 
489  9.87209 
470  9.87198 
451  9.87187 
431  9.87175 

89515  9.95*90 
567  9.95215 
620  9.95240 
672  9.95266 
725  9.95291 

0.04810  I.II7I 
0.04785  165 
0.0476O  158 
0.04734  152 
0.04709  145 

10 

9 
8 

7 
6 

T 

4 
3 

2 

I 

0 

55 

56 

% 

8 

66805  9.82481 
827  9.82495 
848  9.82509 
870  9.82523 
891  9-82537 
913  9-82551 

74412  9.87164 
392  9.87153 

373  9-87141 
353  9.87130 
334  9.87119 
314  9.87107 

89777  9.953I7 
830  9.95342 
883  9.95368 
935  9-95393 
988  9.95418 
90040  9.95444 

0.04683  I.H39 
0.04658  132 
0.04632  126 
0.04607  119 
0.04582  113 

0.04556  106 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

f 

48C 


42° 


f 

Nat.  Sin  Log.  d. 

Nat.  COS  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 
4 

66913  9.82551 
935  9-82565 
956  9.82579 
978  9.82593 
999  9.82607 

14 
14 
H 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
13 
*4 
14 
14 
H 
14 
14 
13 
14 
14 
J4 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
J3 
14 
14 
13 
14 
13 
14 
14 
13 
14 
13 
14 
14 
13 
14 
13 

74314  9.87107 
295  9.87096 
276  9.87085 
256  9-87073 
237  9.87062 

ii 
ii 

12 
II 
12 
II 
II 
12 
II 
12 
II 
12 
II 
12 
II 
12 
II 
II 
12 
II 
12 
12 
II 
12 
II 
12 
II 
12 
II 
12 
II 
12 
12 
II 
12 
II 
12 
12 
II 
12 
12 
II 
12 
12 
II 
12 
12 
II 
12 
12 
12 
II 
12 
12 
II 
12 
12 
12 
II 
12 

90040  9.95444 

°93  9-95469 

146  9.95495 
199  9-95520 
251  9-95545 

11 

25 
25 
26 

% 

25 
25 
26 

25 
25 
26 

25 
26 

25 
25 
26 

25 
26 

25 

% 

25 
25 
26 
25 
26 

25 
25 
26 
25 

2I 
26 

25 
25 
26 

25 
25 
26 

25 
26 

25 
25 
26 

25 
25 
26 

25 
25 
26 
25 
25 
26 

25 

11 

25 

3 

0.04556  1.1106 
0.04531   100 
0.04505   093 
0.04480   087 
0.04455   080 

60 

59 
58 
57 
56 
55 
54 
53 
52 
Si 
50 
49 
48 

47 
46 
45 
44 
43 
42 

4i 
40 

P 
| 

5 

6 

I 

9 

67021  9.82621 
043  9.82635 
064  9.82649 
086  9.82663 
107  9.82677 

74217  9.87050 
198  9.87039 
178  9.87028 
J59  9-87016 
139  9.87005 

90304  9.95571 
357  9-95596 
410  9.95622 

463  9-95647 
516  9.95672 

0.04429  1.1074 
0.04404   067 
0.04378   06  1 

0-04353   °54 
0.04328   048 

10 

ii 

12 
13 
14 

67129  9.82691 
151  9.82705 
172  9.82719 
194  9.82733 
215  9.82747 

74120  9.86993 
loo  9.86982 
080  9.86970 
061  9.86959 
041  9.86947 

90569  9-95698 
621  9-95723 
674  9-95748 
727  9-95774 
781  9.95799 

0.04302  1.1041 
0.04277   035 
0.04252   028 

0.04226    022 

0.04201   016 

15 

16 

17 

18 

19 

67237  9.82761 
258  9.82775 
280  9.82788 
301  9.82802 
323  9.82816 

74022  9.86936 

002  9.86924 

73983  9.86913 
963  9.86902 

944  9.86890 

90834  9.95825 
887  9-95850 
940  9.95875 
993  9-95901 
91046  9.95926 

0.04175  I.I009 
O.04lgO    003 
0.04I2g  1.0996 
0.04099    99° 
0.04074    983 

20 

21 
22 

23 
24 

67344  9.82830 
366  9.82844 
387  9.82858 
409  9.82872 
430  9.82885 

73924  9.86879 
904  9.86867 
885  9.86855 
865  9.86844 
846  9.86832 

91099  9.95952 

153  9-95977 
206  9.96002 
259  9.96028 
313  9.96053 

0.04048  1.0977 
0.04023    971 
0.03998    964 
0.03972    958 

0.03947   951 

25 

26 
27 
28 
29 

67452  9.82899 
473  9.82913 
495  9.82927 
516  9.82941 
538  9-82955 

73826  9.86821 
806  9.86809 
787  9.86798 
767  9.86786 
747  9.86775 

91366  9.96078 
419  9.96104 
473  9.96129 
526  9.96155 
580  9.96180 

0.03922  1.0945 
0.03896   939 
0.03871   932 
0-03845   926 
0.03820   919 

35 

34 
33 
32 
3i 

30 

3i 
32 
33 
34 
35 
36 

P 

39 
40 

41 

42 

43 
44 

67559  9-82968 
580  9.82982 
602  9.82996 
623  9.83010 
645  9.83023 

73728  9.86763 
708  9.86752 
688  9.86740 
669  9.86728 
649  9.86717 

91633  9.96205 
687  9.96231 
740  9.96256 
794  9.96281 
847  9-96307 

0.03795  I'°9I3 
0.03769   907 
0.03744   900 
0.03719   894 
0.03693   888 

30 

I 

67666  9.83037 
688  9.83051 
709  9.83065 
730  9.83078 
752  9.83092 

73629  9.86705 
610  9.86694 

570  9.86670 
551  9.86659 

91901  9.96332 

955  9-96357 
92008  9.96383 
062  9.96408 
116  9.96433 

0.03668  1.0881 
0.03643   875 
0.03617   869 
0.03592   862 
0.03567   856 

25 

24 
23 

22 
21 

67773  9-83106 
795  9-83120 
816  9.83133 
837  9.83147 
859  9.83161 

73531  9.86647 
511  9.86635 
491  9.86624 
472  9.86612 
452  9.86600 

92170  9.96459 
224  9.96484 
277  9-965™ 

0.03541  1.0850 
0.03516   843 
0.03490   837 
0.03465   831 
0.03440   824 

20 

19 
18 

| 

45 

46 

47 
48 

49 

67880  9.83174 
901  9.83188 
923  9.83202 
944  9.83215 
965  9.83229 

73432  9-86589 
413  9.86577 
393  9-86565 
373  9-86554 
353  9.86542 

92439  9.96586 
493  9-966ii 
547  9-96636 
60  1  9.96662 
655  9.96687 

0.03414  1.0818 
0.03389   812 
0-03364   805 
0-03338   799 
0.03313   793 

15 

14 
13 

12 
II 

50 

51 
52 
53 
54 

67987  9.83242 
68008  9.83256 
029  9.83270 
051  9.83283 
072  9.83297 

73333  9.86530 
314  9.86518 
294  9.86507 
274  9.86495 
254  9.86483 

92709  9.96712 
763  9-96738 
817  9.96763 
872  9.96788 
926  9.96814 

0.03288  1.0786 
0.03262   780 
0.03237   774 
0.03212   768 
0.03186   761 

10 

! 

55 

56 
57 
58 

& 

68093  9.83310 
H5  9-83324 
136  9-83338 
157  9-83351 
179  9-83365 

200  9.83378 

73234  9.86472 
215  9.86460 
195  9.86448 
175  9.86436 
155  9.86425 
135  9.86413 

92980  9.96839 
93034  9.96864 
088  9.96890 

143  9-969I5 
197  9.96940 
252  9.96966 

0.03161  1.0755 
0.03136   749 
0.03110   742 
0.03085   736 
0.03060   730 
0-03034   724 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log.  c.d.  Log.Tan  Nat 

r 

47° 


43C 


f 

Nat.  Sin  Log 

d. 

Nat.  COS  Log. 

d. 

Nat.  Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

I 

2 

3 

4 

68200  9.83378 

221  9.83392 
242  9.83405* 
264  9.83419 
285  9.83432 

14 
13 
14 
13 

73135  9.86413 
116  9.86401 
096  9.86389 
076  9.86377 
056  9.86366 

12 
12 
12 
II 

93252  9.96966 
306  9.96991 

360  9.97016 

415  9-97042 
469  9.97067 

25 

11 

25 

0.03034  1.0724 
0.03009   717 
0.02984   711 
0.02958   705 
0.02933   699 

60 

59 
58 
57 
56 

5 

6 

I 

9 

68306  9.83446 
327  9.83459 

349  9.83473 
370  9.83486 
391  9.83500 

4  H  H  H  M  h 
)  4*.  CO  4^.  CO  4: 

73036  9.86354 
016  9.86342 
72996  9.86330 
976  9.86318 
957  9-86306 

12 
12 
12 
12 

93524  9.97092 
578  9.97118 
633  9-97143 

688  9.97168 
742  9-97I93 

26 
25 
25 
25 
26 

0.02908  1.0692 
0.02882   686 
0.02857   680 
0.02832   674 
0.02807   668 

55 

54 
53 
52 

10 

II 

12 

13 

14 

68412  9.83513 
434  9-83527 
455  9-83540 
476  9-83554 
497  9-83567 

-  Co  4-  Co  4-  U 

72937  9.86295 
917  9.86283 
897  9.86271 
877  9.86259 
857  9.86247 

12 
12 
12 
12 

93797  9-97219 
852  9.97244 
906  9.97269 
961  9.97295 
94016  9.97320 

25 
25 

0.02781  1.0661 
0.02756   655 
0.02731   649 
0.02705   643 
0.02680   637 

50 

49 
48 
47 
46 

15 
16 

19 

68518  9.83581 

539  9-83594 
561  9.83608 
582  9.83621 
603  9.83634 

*•  CO^COCO  ' 

H  H  M  H  M  h 

72837  9.86235 
817  9.86223 
797  9.86211 
777  9.86200 
757  9.86188 

12 
12 
II 
12 

94071  9-97345 
125  9-97371 
180  9.97396 
235  9-97421 
290  9-97447 

26 
25 
25 
26 

0.02655  1.0630 
0.02629   624 
0.02604   618 
0.02579   612 
0-02553   606 

45 

44 
43 
42 

20 

21 

22 

23 
24 

68624  9.83648 
645  9.83661 
666  9.83674 
688  9.83688 
709  9.83701 

•"••t 

13 
14 
13 

72737  9.86176 
717  9.86164 
697  9.86152 
677  9.86140 
657  9.86128 

12 
12 
12 
12 

94345  9-97472 
400  9.97497 
455  9*975^3 

5io  9.97548 

565  9-97573 

25 
25 

0.02528  1.0599 
0.02503   593 
0.02477   587 
0.02452   581 
0.02427   575 

40 

P 
1 

25 

26 

27 
28 

29 

68730  9.83715 
751  9-83728 
772  9.83741 

793  9-83755 
814  9.83768 

;*•  co  co  rf-  co  ( 

H  hH  M  M  H  h 

72637  9.86116 
617  9.86104 
597  9.86092 
577  9.86080 
557  9.86068 

12 
12 
12 
12 

94620  9.97598 
676  9.97624 
731  9.97649 
786  9.97674 
841  9.97700 

25 
26 
25 

0.02402  1.0569 
0.02376   562 
0.02351   556 
0.02326   550 
0.02300   544 

35 

34 
33 
32 

30 

32 
33 
34 

68835  9.83781 
857  9.83795 
878  9.83808 
899  9.83821 
920  9.83834 

AJ 
14 
13 
13 
13 

72537  9-86056 
517  9.86044 
497  9.86032 
477  9.86020 
457  9.86008 

12 
12 
12 
12 

94896  9.97725 
952  9-97750 
95007  9.97776 
062  9.97801 
118  9.97826 

25 

11 

25 
25 

0.02275  1.0538 
0.02250   532 
0.02224   526 
0.02199   519 
0.02174   513 

30 

27 
26 

35 

36 

39 

68941  9.83848 
962  9.83861 

983  9.83874 
69004  9.83887 
025  9.83901 

j  4*  CO  CO  CO  4 

72437  9.85996 
417  9-85984 
397  9.85972 
377  9-85960 
357  9-85948 

12 
12 
12 
12 

95J73  9.97851 
229  9.97877 
284  9.97902 
340  9.97927 
395  9-97953 

25 
26 
25 

11 

0.02149  1.0507 
0.02123   501 
0.02098   495 
0.02073   489 
0.02047   483 

25 

24 
23 

22 
21 

40 

42 
43 
44 

69046  9.83914 
067  9.83927 
088  9.83940 
109  9.83954 

O  COCO  TtCO  C 

4  M  IH  M  H  t- 

72337  9-85936 
317  9-85924 
297  985912 
277  9.85900 
257  9.85888 

12 
12 
12 
12 

95451  9-97978 
506  9.98003 
562  9.98029 
618  9.98054 
673  9.98079 

"O 

25 
26 

25 
25 

0.02022  1.0477 
0.01997   470 
0.01971   464 
0.01946   458 
0.01921   452 

20 

19 
18 

17 
16 

45 

46 

49 

69151  9.83980 
172  9.83993 
193  9.84006 
214  9.84020 
235  9.84033 

•••o 

13 
13 
14 

72236  9.85876 
216  9.85864 
196  9.85851 
176  9-85839 
156  9.85827 

12 

13 
12 
12 

95729  9-98104 
785  9.98130 
841  9.98155 
897  9.98180 
952  9.98206 

25 
26 
25 
25 
26 

0.01896  1.0446 
0.01870   440 
0.01845   434 
0.01820   428 
0.01794   422 

15 

14 
13 

12 
II 

50 

S2 
53 
54 

69256  9.84046 
277  9.84059 
298  9.84072 
319  9.84085 
340  9.84098 

O  CO  COCO  CO  ' 

^  M  l-l  M  M  H 

72136  9.85815 
116  9.85803 
095  9-8579I 
°75  9-85779 
°55  9.85766 

12 
12 
12 
13 

96008  9.98231 
064  9.98256 

120  9.98281 
176  9.98307 
232  9.98332 

25 

25 

25 

0.01769  1.0416 
0.01744   410 
0.01719   404 
0.01693   398 
0.01668   392 

10 
I 

55 

56 

11 
i'o 

69361  9.84112 
382  9.84125 
403  9.84138 
424  9.84151 
445  9.84164 
466  9.84177 

CO  CO  CO  CO  CO  4 

72035  9.85754 
015  9.85742 
71995  9-85730 
974  9-85718 
954  9-85706 
934  9-85693 

12 
12 
12 
12 
13 

96288  9.98357 

344  9-98383 
400  9.98408 

457  9-98433 
513  9.98458 
569  9.98484 

26 
25 
25 

0.01643  1.0385 
0.01617   379 
0.01592   373 
0.01567   367 
0.01542   361 
0.01516   355 

5 

4 
3 

2 

I 

0 

Nat.  COS  Log 

.  d. 

Nat.  Sin  Log. 

d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

t 

46C 


44C 


f 

Nat.  Sin  Log 

d. 

Nat.  COS  Log 

d. 

Nat.TanLog 

c.d 

Log.  Cot  Nat 

0 

I 

2 

3 

4 

69466  9.84177 
487  9.84190 
508  9.84203 
529  9.84216 
549  9.84229 

J  CO  CO  CO  CO 

71934  9.85693 
914  9.85681 
894  9.85669 
873  9.85657 
853  9-85645 

12 
12 
12 
12 

96569  9.98484 
625  9.98509 

68  1  9.98534 
738  9.98560 
794  9-98585 

25 

25 
26 

25 

0.01516  1.0355 
0.01491   349 
0.01466   343 
0.01440   337 
0.01415   331 

60 

it 
1 

5 

6 

.9 

69570  9.84242 

633  9.84282 
654  9.84295 

o  co  TJ-  co  co  c 

71833  9.85632 
813  9.85620 
792  9.85608 

752  9-85583 

I3 

12 
12 
12 

96850  9.98610 

907  9.98635 
963  9.98661 
97020  9.98686 
076  9.98711 

25 

25 
25 

0.01390  1.0325 
0.01365   319 

0-01339   313 
0.01314   307 
0.01289   301 

55 

54 
53 
52 

10 

ii 

12 
13 

14 

69675  9.84308 
696  9.84321 
717  9.84334 
737  9.84347 
758  9.84360 

O  CO  CO  CO  CO  C 
H  M  M  H  H  H 

71732  9.85571 
711  9-85559 
691  9-85547 
671  9.85534 
650  9-85522 

12 
12 

13 
12 

97133  9.98737 

189  9.98762 

246  9.98787 
302  9.98812 

359  9.98832 

25 
25 
25 
26 

0.01263  I-°295 
0.01238   289 
0.01213   283 
0.01188   277 
0.01162   271 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

800  9.84385 
821  9.84398 
842  9.84411 
862  9.84424 

ON  rococo  c 

4  H  M  tH  M  H 

71630  9.85510 
610  9.85497 
590  9.85485" 
569  9.85473 
549  9.85460 

13 
12 
12 
13 

97416  9.98863 
472  9.98888 
529  9.98913 
586  9.98939 
643  9.98964 

25 

25 

11 

25 

0.01137  1.0265 

O.OIII2    259 
O.OI087    253 

0.01061   247 
0.01036   241 

45 

44 
43 

42 

20 

21 

22 

23 
24 

69883  9.84437 
904  9.84450 
925  9.84463 
946  9.84476 
966  9.84489 

*a 
13 
13 
13 
13 

71529  9.85448 
508  9-85436 
488  9-85423 
468  9.85411 
447  9.85399 

12 

13 
12 
12 

97700  9.98989 
756  9.99015 
813  9.99040 
870  9.99065 
927  9.99090 

25 
26 
25 
25 
25 

o.oion  1.0235 

0.00985  230 

0.00960   224 
0.00935   218 

0.00910    212 

40 
P 
1 

25 

26 

27 

28 

29 

69987  9.84502 
70008  9.84515 
029  9.84528 
049  9.84540 
070  9.84553 

J  CO  10  CO  CO  O 

71427  9.85386 

407  9-85374 
386  9.85361 
366  9-85349 
345  9-85337 

I3 
12 

13 
12 
12 

97984  9.99116 
98041  9.99141 
098  9.99166 

155  9.99I9I 
213  9.99217 

25 
25 

0.00884  1.  0206 

0.00859    200 

0.00834  194 

0.00809   l88 

0.00783   182 

35 

34 
33 
32 
3i 

30 

32 
33 
34 

70091  9.84566 

112  9.84579 
I32  9.84592 
153  9.84605 
174  9.84618 

0  CO  CO  CO  CO  O 

71325  9.85324 
305  9-853I2 
284  9.85299 
264  9.85287 
243  9-85274 

12 

13 
12 

98270  9.99242 
327  9.99267 
384  9.99293 
441  9.99318 
499  9.99343 

25 
25 
26 

25 
25 

0.00758  1.0176 

0.00733  170 

0.00707   164 
0.00682   158 
0.00657   152 

30 

27 
26 

35 

36 

% 

39 

70195  9.84630 
277  9.84682 

0  CO  CO  CO  CO 

71223  9.85262 
203  9.85250 
182  9.85237 
162  9.85225 
141  9.85212 

12 

13 
12 

98556  9.99368 
613  9.99394 
671  9.99419 
728  9.99444 
786  9.99469 

26 
25 
25 
25 
26 

0.00632  1.0147 
0.00606   141 
0.00581   135 
0.00556  129 
0.00531   123 

25 

24 
23 

22 
21 

40 

41 

42 
43 
44 

70298  9.84694 
319  9.84707 

339  9-84720 
360  9.84733 
381  9.84745 

13 
13 
13 

12 

71121  9.85200 
loo  9.85187 
080  9.85175 
059  9.85162 
039  9-85I50 

13 
12 

13 
12 

98843  9-99495 
901  9.99520 

958  9-99545 
99016  9.99570 
073  9-99596 

25 
25 
25 
26 

0.00505  1.0117 
0.00480   III 

0.00455  105 

0.00430   099 
0.00404   094 

20 

19 
18 

17 
16 

45 

46 

47 
48 

49 

70401  9.84758 
422  9.84771 

463  9.84796 
484  9.84809 

0  CO  CON  CO  C 

71019  9.85137 
70998  9.85125 
978  9.85112 
957  9.85100 
937  9-85087 

12 

13 
12 

13 

99131  9.99621 
189  9.99646 
247  9.99672 
304  9.99697 
362  9.99722 

25 
25 

0.00379  i.  008  8 
0.00354   082 
0.00328   076 
0.00303   070 
0.00278   064 

15 

14 
13 

12 
II 

50 

51 
S2 
53 
54 

70505  9.84822 
525  9.84835 
546  9.84847 
567  9.84860 
587  9.84873 

•3 

13 
12 

13 
13 

70916  9.85074 
896  9.85062 
875  9.85049 
855  9.85037 
834  9.85024 

12 

13 
12 

13 

99420  9.99747 
478  9.99773 
536  9.99798 
594  9.99823 
652  9.99848 

26 
25 
25 
25 

0.00253  1.0058 
0.00227   052 

0.00202    047 
0.00177    041 
0.00152    035 

10 

9 
8 

76 

55 

56 
57 
58 

70608  9.84885 
628  9.84898 
649  9.84911 
670  9.84923 
690  9.84936 
711  9.84949 

CO  CO  K)  CO  CO  tv 

70813  9.85012 

.  793  9.84999 
772  9.84986 
752  9.84974 
731  9.84961 
711  9.84949 

10  CO  lOCOCO  K 

99710  9.99874 
768  9.99899 
826  9.99924 
884  9-99949 
942  9-99975 
IOOOOD  o.ooooo 

25 
25 

% 

25 

0.00126  1.0029 

o.ooioi   023 
0.00076   017 

0.00051    012 
0.00025    006 
0.00000    OOO 

5 

4 

3 

2 

I 

0 

Nat.  COS  Log 

d. 

Nat.  Sin  Log. 

d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

t 

45( 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


JAN  3  0  1948 


RLO 


REC'D  LD 

OCT251957 

, 


NOV  2  1  1958 


CO 


LD  21-100m-9,'47(A5702sl6)476 


YC  22299 


L9 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


